diff --git "a/samples/texts_merged/6422547.md" "b/samples/texts_merged/6422547.md" new file mode 100644--- /dev/null +++ "b/samples/texts_merged/6422547.md" @@ -0,0 +1,4604 @@ + +---PAGE_BREAK--- + +# DIFFERENTIAL MODELS FOR THE ANDERSON DUAL +TO BORDISM THEORIES AND INVERTIBLE QFT'S + +MAYUKO YAMASHITA AND KAZUYA YONEKURA + +ABSTRACT. In this paper, we construct new models for the Anderson duals ($I\Omega_G^G$)* to the stable tangential *G*-bordism theories and their differential extensions. The cohomology theory ($I\Omega_G^G$)* is conjectured by Freed and Hopkins [FH21] to classify deformation classes of possibly non-topological invertible quantum field theories (QFT's). Our model is made by abstractizing certain properties of invertible QFT's, thus supporting their conjecture. + +## CONTENTS + +
1. Introduction2
1.1. A sketch of the main results3
1.2. Physical significance5
1.3. Mathematical significance10
1.4. The structure of the paper10
1.5. Notations11
2. Preliminaries12
2.1. Spectra and generalized cohomology theories12
2.2. The Anderson duals15
2.3. Madsen-Tillmann spectra and stable tangential G-bordism theories17
2.4. Singular (co)homology21
2.5. Generalized differential cohomology theories25
2.6. ⟨k⟩-manifolds27
3. Geometric stable tangential G-structures30
3.1. Geometric stable tangential G-structures30
3.2. Geometric Pontryagin-Thom construction34
4. Physically motivated models for the Anderson duals of G-bordisms38
4.1. The models39
4.2. Examples of elements in (IΩdRG)*58
4.3. The proof that (IΩsingG)* is a generalized cohomology theory62
4.4. Self-duality homomorphisms74
5. The proof of the main results76
5.1. The proof of IZsing* ≃ IZ*76
+ +RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, KYOTO UNIVERSITY, 606-8502, KYOTO, JAPAN + +DEPARTMENT OF PHYSICS, TOHOKU UNIVERSITY, SENDAI 980-8578, JAPAN + +E-mail addresses: mayuko@kurims.kyoto-u.ac.jp, yonekura@tohoku.ac.jp. +---PAGE_BREAK--- + +5.2. The proof of $(I\Omega_{\text{sing}}^G)^* \simeq (I\Omega^G)^*$ 82 + +6. Examples revisited 102 + +7. Concluding remarks 105 + +7.1. Normal $G$-structures 105 + +7.2. Module structures 107 + +Acknowledgment 107 + +References 107 + +# 1. INTRODUCTION + +In this paper, we construct new models for the Anderson duals $(I\Omega^G)^*$ to the stable tangential $G$-bordism theories and their differential extensions. Freed and Hopkins [FH21] conjectured that the generalized cohomology theory $(I\Omega^G)^*$ classifies deformation classes of possibly non-topological invertible quantum field theories (QFT's) on stable tangential $G$-manifolds. Our model is physically motivated, in the sense that it is made by abstractizing certain properties of invertible QFT's. Thus the result of this paper supports their conjecture. + +Associated to a generalized cohomology theory $E^*$, its *Anderson dual* ([HS05, Appendix B], [FMS07, Appendix B]) is a generalized cohomology theory which we denote by $IE^*$. The crucial property of this theory is that it fits into the following exact sequence for any CW-complex $X$. + +$$ +\begin{equation} \tag{1.1} +\begin{aligned} +& \cdots \to \operatorname{Hom}(E_{n-1}(X), \mathbb{R}) \to \operatorname{Hom}(E_{n-1}(X), \mathbb{R}/\mathbb{Z}) \to I\mathrm{E}^n(X) \\ +& \qquad \to \operatorname{Hom}(E_n(X), \mathbb{R}) \to \operatorname{Hom}(E_n(X), \mathbb{R}/\mathbb{Z}) \to \cdots \text{ (exact).} +\end{aligned} +\end{equation} +$$ + +In this paper we are interested in the Anderson dual to *stable tangential G-bordism theories* $\Omega^G$. Here $G = \{G_d, s_d, \rho_d\}_{d \in \mathbb{Z}_{\ge 0}}$ is a sequence of compact Lie groups equipped with homomorphisms $s_d: G_d \to G_{d+1}$ and $\rho_d: G_d \to O(d, \mathbb{R})$ for each $d$ which are compatible with the inclusion $O(d, \mathbb{R}) \hookrightarrow O(d+1, \mathbb{R})$. The homology theory corresponding to $\Omega^G$ is given by the stable tangential $G$-bordism groups $\Omega^G_*(X)$, and the exact sequence (1.1) becomes + +$$ +\begin{equation} +\begin{split} +& \cdots \to \operatorname{Hom}(\Omega_{n-1}^G(X), \mathbb{R}) \to \operatorname{Hom}(\Omega_{n-1}^G(X), \mathbb{R}/\mathbb{Z}) \to (I\Omega^G)^n(X) \\ +& \qquad \to \operatorname{Hom}(\Omega_n^G(X), \mathbb{R}) \to \operatorname{Hom}(\Omega_n^G(X), \mathbb{R}/\mathbb{Z}) \to \cdots (\text{exact}). +\end{split} +\tag{1.2} +\end{equation} +$$ + +The starting point of this work is the following conjecture by Freed and Hopkins. + +**Conjecture 1.3** ([FH21, Conjecture 8.37]). There is a 1 : 1 correspondence¹ + +$$ +(1.4) \quad \left\{ \text{deformation classes of reflection positive invertible } n\text{-dimensional extended field theories with symmetry type } G \right\} \simeq (I\Omega^G)^{n+1}(\mathrm{pt}). +$$ + +¹Here *symmetry types* of QFT’s in [FH21] are certain classes of *G’s* in this paper which satisfy an additional set of conditions. +---PAGE_BREAK--- + +Difficult points of Conjecture 1.3 are the following. First, we do not have the axioms for non-topological QFT’s. Thus the left hand side of (1.4) is not a mathematically well-defined object. Second, although the cohomology theory $(I\Omega^G)^*$ is mathematically defined, its definition is abstract. So the right hand side of (1.4) is difficult to treat directly. Actually, those difficulties are overcome if we are interested in topological QFT’s, and Freed and Hopkins proves the version of Conjecture 1.3 for topological QFT’s, where the right hand side of (1.4) is replaced by its torsion part [FH21, Theorem 1.1]. + +This work is intended to overcome the second difficulty mentioned above, and to give a new approach to Conjecture 1.3. We construct a physically motivated model for the theory $(I\Omega^G)^*$, which is made by abstractizing certain properties of invertible QFT’s. This result supports Conjecture 1.3. On the other hand, our results also turn out to be mathematically interesting, in view of its relations to *differential cohomology theories*. + +In the rest of the introduction, we first explain the main result of this paper in Subsection 1.1, and then explain its physical significance in Subsection 1.2, and its mathematical significance in Subsection 1.3. + +**1.1. A sketch of the main results.** The main results of this paper is the construction of models for the generalized cohomology theory $(I\Omega^G)^*$ and its differential extension. Actually we construct two models, the *differential model* $(I\Omega_{\mathrm{dR}}^G)^*$ (Definition 4.11) which is defined for manifolds, and the *singular model* $(I\Omega_{\mathrm{sing}}^G)^*$ (Definition 4.56) which is defined for topological spaces. Here we only sketch the construction of the differential model. The precise definition is given in Definition 4.11. The physical meaning of this construction is explained in Subsection 1.2 below. For simplicity, in this subsection we only consider the case where $G$ is oriented, i.e., the image of $\rho_d: G_d \to \mathrm{O}(d, \mathbb{R})$ lies in $\mathrm{SO}(d, \mathbb{R})$ for each $d$. + +Let $X$ be a manifold and let $n$ be a nonnegative integer. The construction starts by defining the larger group $(I\Omega_{\mathrm{dR}}^G)^n(X)$, which is going to be the model for the differential extension. It consists of pairs $(\omega, h)$, where + +* $\omega \in \Omega_{\mathrm{clo}}^n(X; \lim_{\leftarrow d} (\mathrm{Sym}^{\bullet/2} g_d^*))$, i.e., $\omega$ is a closed differential form on $X$ with values in invariant polynomials on $g := \lim_{\leftarrow d} g_d$, of total degree $n$, where $g_d$ is the Lie algebra of $G_d$. + +* $h$ is a map which assigns an $\mathbb{R}/\mathbb{Z}$-value to a triple $(M, g, f)$, where $M$ is a closed $(n-1)$-dimensional manifold with a stable tangential $G$-structure with connection, which we call a *geometric stable tangential $G$-structure* and symbolically denoted by $g$, and a smooth map $f: M \to X$. + +* $\omega$ and $h$ should satisfy the following compatibility condition. Suppose we have a compact $n$-dimensional manifold with boundary $(W, \partial W)$ equipped with a geometric stable tangential $G$-structure $g_W$ and a map $f_W: W \to X$. Assume that all the structures are compatible with a collar structure near $\partial W$, so that it defines a triple $(\partial W, \partial g_W, f_W|_{\partial W})$ as above. The data $g_W$ allows us to define +---PAGE_BREAK--- + +a top form on $W$, + +$$ (1.5) \qquad \mathrm{cw}_{g_W}(f_W^*\omega) \in \Omega^n(W), $$ + +by applying the Chern-Weil construction with respect to $g_W$ to the coefficient of $f_W^*\omega$. We require that, + +$$ (1.6) \qquad h([\partial W, \partial g_W, f_W | \partial W]) = \int_W \mathrm{cw}_{g_W}(f_W^*\omega) \pmod{\mathbb{Z}}. $$ + +To define $(I\Omega_{\mathrm{dR}}^G)^n(X)$, we introduce the equivalence relation $\sim$ on $(\widehat{I\Omega_{\mathrm{dR}}^G})^n(X)$. We set $(\omega, h) \sim (\omega', h')$ if there exists $\alpha \in \Omega^{n-1}(X; \lim_{\leftarrow d} (\mathrm{Sym}^{\bullet/2} g_d^*))$ such that + +$$ \omega' = \omega + d\alpha, $$ + +$$ (1.7) \qquad h'([M, g, f]) = h([M, g, f]) + \int_M \mathrm{cw}_g(f^*\alpha). $$ + +We define + +$$ (1.8) \qquad (I\Omega_{\mathrm{dR}}^G)^n(X) := (\widehat{I\Omega_{\mathrm{dR}}^G})^n(X) / \sim. $$ + +The main result of this paper concerning the differential model is the following. + +**Theorem 1.9** (Proposition 4.70, Proposition 4.122 and Theorem 5.21). $(I\Omega_{\mathrm{dR}}^G)^*$ gives a model for the generalized cohomology theory $(I\Omega^G)^*$, restricted to the category of manifolds. Moreover, $(\widehat{I\Omega_{\mathrm{dR}}^G})^*$ is a model for the differential extension of $(I\Omega^G)^*.$ + +To conclude this subsection, we give an easiest example of elements in $(\widehat{I\Omega_{\mathrm{dR}}^G})^*$. For more examples, see Subsection 4.2. + +*Example 1.10* (The holonomy theory (1), Example 4.79). In this example we consider $G = \mathrm{SO} = \{\mathrm{SO}(d, \mathbb{R})\}_d$. Fix a manifold $X$ and a hermitian line bundle with unitary connection $(L, \nabla)$ over $X$. Then we get an element + +$$ (c_1(\nabla), \mathrm{Hol}_{\nabla}) \in (\widehat{I\Omega_{\mathrm{dR}}^{\mathrm{SO}}})^2(X). $$ + +Here, + +• $c_1(\nabla) = \frac{\sqrt{-1}}{2\pi} F_\nabla \in \Omega_{\mathrm{clo}}^2(X)$ is the first Chern form of $\nabla$. + +• Note that a triple $(M, g, f)$ in this case consists of a closed oriented 1-dimensional manifold $M$ with a map $f: M \to X$, i.e., a closed oriented curve in $X$, along with other data. The map $\mathrm{Hol}_\nabla$ assigns the holonomy of $(L, \nabla)$ along the curve, using the identification $U(1) \simeq \mathbb{R}/\mathbb{Z}$. + +Then, the compatibility condition (1.6) follows from the relation of curvature and holonomy, namely for a compact oriented 2-dimensional manifold $(W, \partial W)$ with a map $f_W: W \to X$, we have + +$$ \mathrm{Hol}_{\nabla}(f_W|_{\partial W}) = \int_W f_W^* c_1(\nabla) \pmod{\mathbb{Z}}. $$ +---PAGE_BREAK--- + +1.2. **Physical significance.** Our results are motivated by the problem of classification of invertible field theories. Let us explain some background in physics. In the following discussion, whenever we say “manifolds”, they are always supposed to be equipped with some geometric structure such as Riemannian metric, bundles and their connections, and so on. What geometric structure we consider should be specified in advance. + +Very roughly speaking, a $D$-dimensional QFT is a functor from some geometric bordism category to the (super)vector space category as follows. A QFT assigns a Hilbert space of physical states $\mathcal{H}(N)$ to each $(D-1)$-dimensional closed manifold $N$. In particular, we assume that for the empty manifold $N=\emptyset$, we have a canonical isomorphism $\mathcal{H}(\emptyset) \simeq \mathbb{C}$. It assigns a linear map $Z(M) : \mathcal{H}(N_1) \to \mathcal{H}(N_2)$ to each $D$-dimensional compact manifold $M$ with boundaries $\partial M = \bar{N}_1 \cup N_2$ where $\bar{N}_1$ is a manifold which has the opposite structure to that of $N_1$ (such as orientation reversal), and we have assumed that $M$ has appropriate collar structure near the boundaries, $[0, \epsilon) \times N_1$ and $(-\epsilon, 0] \times N_2$ for some $\epsilon > 0$. We do not try to make these axioms precise, but we remark that they are motivated by (Euclidean) path integrals in physics. + +An invertible field theory is a QFT in which the Hilbert space of states $\mathcal{H}(N)$ on any closed manifold $N$ is one-dimensional, $\dim \mathcal{H}(N) = 1$. Invertible field theories play crucial roles in the study of anomalies. (See e.g. [Fre14, Mon19] for overviews.) In fact, the classification of deformation classes of invertible QFT's in $D$-dimensions is believed to be the same as classification of anomalies in $(D-1)$-dimensions.² (We will explain what we mean by “deformation classes” in a little more detail later.) In the context of condensed matter physics, deformation classes of invertible field theories are also called symmetry protected topological (SPT) phases or invertible phases of matter. Anomalous ($D-1$)-dimensional theories appear on the boundaries of these invertible phases and have various applications in physics. Therefore, it is an important problem to classify invertible phases. + +In the case of topological QFT (TQFT), the classification of invertible phases has been conjectured to be given by certain cobordism groups [Kap14, KTTW14], and later proved at least for some physically motivated classes of structure types and under some axioms of TQFT [FH21, Yon18]. Let $S$ be the structure type under consideration. For instance, we can consider manifolds equipped with Spin structures, and in that case we denote $S = \text{Spin}$. + +Then we may define a bordism group $\Omega_D^S(\text{pt})$ of $D$-dimensional manifolds equipped with structure of the type $S$ roughly as follows. We introduce a monoid structure on the set of (isomorphism classes of) manifolds by disjoint union, $M_1 \sqcup M_2$. The empty manifold $\emptyset$ is the unit of this monoid since $M \sqcup \emptyset \simeq M$. Then we divide this monoid by an equivalence relation. If a closed $D$-manifold $M$ is a boundary of some $(D+1)$-manifold $W$, $M = \partial W$, then it is defined to be equivalent to the empty set, $M \sim \emptyset$. By using the fact that $W = [0,1] \times M$ has the boundary $\partial W = M \sqcup \bar{M}$, one can see that we get a group whose elements are represented in terms of manifolds + +²We neglect anomalies which do not fit into the general framework, such as Weyl anomalies. Also, there may be subtleties in reflection non-positive theories [CL20]. +---PAGE_BREAK--- + +$M$ as $[M]$. In particular, the inverse of $[M]$ is $\overline{[M]}$. This group is denoted as $\Omega_D^S(\text{pt})$. + +According to [Kap14, KTTW14, FH21, Yon18], deformation classes of invertible TQFT's are classified by the group $\operatorname{Hom}((\Omega_D^S(\text{pt}))_{\text{tor}}, \mathbb{R}/\mathbb{Z})$, where the subscript tor means to take the torsion part of the group. The reason that we take the torsion part is that we are considering deformation classes. To explain this point, let us first consider the group $\operatorname{Hom}(\Omega_D^S(\text{pt}), \mathbb{R}/\mathbb{Z})$. Then the relation between this group and the above axioms of QFT is the following. If we are given an element $h \in \operatorname{Hom}(\Omega_D^S(\text{pt}), \mathbb{R}/\mathbb{Z})$, it means that we can assign to each closed $D$-dimensional manifold $M$ a number + +$$ (1.11) \qquad Z(M) = \exp(2\pi\sqrt{-1}h([M))), $$ + +where $[M] \in \Omega_D^S(\text{pt})$ is the bordism class represented by $M$. Notice that for a closed manifold $\partial M = \emptyset$, a QFT should assign a linear map $Z(M) : \mathcal{H}(\emptyset) \to \mathcal{H}(\emptyset)$. Since $\mathcal{H}(\emptyset) \simeq \mathbb{C}$, the quantity $Z(M)$ can be regarded just as a number $Z(M) \in \mathbb{C}$. The function which assigns a number $Z(M) \in \mathbb{C}$ to each closed manifold $M$ is called a partition function in physics. From an element $h \in \operatorname{Hom}(\Omega_D^S(\text{pt}), \mathbb{R}/\mathbb{Z})$, we can construct a partition function by (1.11). + +A partition function itself does not give full data for the axioms of QFT. However, the theorems proved in [FH21, Yon18] imply that we can construct a TQFT from a given $h \in \operatorname{Hom}(\Omega_D^S(\text{pt}), \mathbb{R}/\mathbb{Z})$ (see Theorem 4.3 of [Yon18] for explicit construction), and also the partition function of any invertible TQFT can be deformed continuously to a partition function given by some $h \in \operatorname{Hom}(\Omega_D^S(\text{pt}), \mathbb{R}/\mathbb{Z})$. Among the elements of $\operatorname{Hom}(\Omega_D^S(\text{pt}), \mathbb{R}/\mathbb{Z})$, the ones which come from $\operatorname{Hom}(\Omega_D^S(\text{pt}), \mathbb{R})$ can be deformed continuously. In this way, we arrive at the classification of deformation classes of invertible TQFT's by $\operatorname{Hom}((\Omega_D^S(\text{pt}))_{\text{tor}}, \mathbb{R}/\mathbb{Z})$. + +How about the cases which are not necessarily topological? Freed and Hopkins have conjectured a classification [FH21] in terms of the Anderson dual of bordism groups as stated in Conjecture 1.3. Before going to discuss this conjecture, let us first explain some additional background. + +Suppose we are given a manifold $X$. Then we can consider a new structure given as follows. In addition to the geometric structure already present in a manifold $M$, we consider an additional datum $f: M \to X$ which is a map from $M$ to $X$. In the context of invertible phases and anomalies in physics, we can consider various types of $X$. If the manifold $X$ is taken to be the target space of a sigma model, it is relevant to sigma model anomalies [MN84, MN85, Tho17]. On the other hand, if $X$ is taken to be the space of coupling constants, it is relevant to more subtle anomalies discussed in e.g. [TY17, STY18, ST19, LMSN18, CFLS19a, CFLS19b, HKT20]. Let us denote the new structure type as $(S, X)$, where $S$ is the original one already considered on $M$. Then we denote $\Omega_D^S(X) := \Omega_D^{(S,X)}(\text{pt})$. For appropriate structure types $S$, it is known that $\Omega_*^S$ gives a generalized homology theory, and $\Omega_*^S(X)$ are the generalized homology groups of $X$. + +Given a generalized homology theory $E_*$, we have the Anderson dual cohomology theory $IE^*$ satisfying the exact sequence (1.1). This exact sequence +---PAGE_BREAK--- + +is analogous to the one in the ordinary cohomology theory associated to the short exact sequence of coefficient groups $0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z} \to 0$. + +The conjecture of Freed-Hopkins, with a generalization including sigma models and more general types of $\mathcal{S}$, is that deformation classes of invertible field theories with the structure type $(\mathcal{S}, X)$ is given by $(I\Omega^S)^{D+1}(X)$, where $(I\Omega^S)^*$ is the Anderson dual of $\Omega_*^S$. It fits into the exact sequence + +$$ +\begin{align*} +& \dots \to \operatorname{Hom}(\Omega_D^S(X), \mathbb{R}) \to \operatorname{Hom}(\Omega_D^S(X), \mathbb{R}/\mathbb{Z}) \to (I\Omega^S)^{D+1}(X) \\ +& \qquad \to \operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R}) \to \operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R}/\mathbb{Z}) \to \dots +\end{align*} +$$ + +Let us explain physical reasons to believe that this conjecture is reasonable, following [LOT20]. (See also [DL20] for some applications.) + +First, let us consider elements of $(I\Omega^S)^{D+1}(X)$ which are in the kernel of the map $(I\Omega^S)^{D+1}(X) \to \operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R})$. We denote the homomorphism $\operatorname{Hom}(\Omega_D^S(X), \mathbb{R}) \to \operatorname{Hom}(\Omega_D^S(X), \mathbb{R}/\mathbb{Z})$ as $p$. By the exact sequence, the kernel is isomorphic to + +$$ +(1.12) \quad \operatorname{Hom}(\Omega_D^S(X), \mathbb{R}/\mathbb{Z}) / \operatorname{Im}(p) \simeq \operatorname{Hom}((\Omega_D^S(pt))_{\operatorname{tor}}, \mathbb{R}/\mathbb{Z}). +$$ + +This is what we have discussed before in the case of TQFT. Any element of $\operatorname{Hom}(\Omega_D^S(X), \mathbb{R}/\mathbb{Z})$ gives a TQFT. The division by the image of the map $p: \operatorname{Hom}(\Omega_D^S(X), \mathbb{R}) \to \operatorname{Hom}(\Omega_D^S(X), \mathbb{R}/\mathbb{Z})$ is due to the fact that we are considering deformation classes. The group $\operatorname{Hom}(\Omega_D^S(X), \mathbb{R})$ is a vector space over $\mathbb{R}$, and any two elements of this group can be continuously deformed into one another. Therefore, we should divide $\operatorname{Hom}(\Omega_D^S(X), \mathbb{R}/\mathbb{Z})$ by $\operatorname{Im}(p)$ when we consider deformation classes of TQFT's. + +Next, let us consider the physical meaning of the map $(I\Omega^S)^{D+1}(X) \to \operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R})$. In physics, we may expect the following property of invertible QFT. Suppose that a closed $D$-manifold $M$ is the boundary of a ($D+1$)-manifold $W$ with a collar structure $(-\epsilon, 0] \times M \subset W$ near the boundary, including geometric data such that $(-\epsilon, 0]$ has the trivial geometric structure. We expect to have a closed differential ($D+1$)-form $I_{D+1}$ on $W$ (which is sometimes called an anomaly polynomial in the context of anomalies). It is constructed from geometric data on $W$. For example, $W$ may have connections of some bundles from which we can construct characteristic forms. Also, $W$ is equipped with a map $f_W: W \to X$ and hence we can pullback differential forms from $X$ to $W$ by using $f_W$. The closed form $I_{D+1}$ is constructed by using such differential forms, and it is given by $cw_{g_W}(f_W^*\omega)$ which appeared in (1.5) in the case of $\mathcal{S}=G$. Then, physicists may expect that the partition function of an invertible QFT evaluated on $M=\partial W$ is given (after some continuous deformation of the theory$^3$) by + +$$ +(1.13) \qquad Z(\partial W) = \exp \left( 2\pi \sqrt{-1} \int_W I_{D+1} \right). +$$ + +³In generic theories, there can be nonuniversal terms such as the cosmological constant of the background Riemannian metric and the Euler number term. We need to eliminate them by continuous deformation of the theory for the following claim to be valid. This deformation can be done by a procedure similar to (1.15) below. +---PAGE_BREAK--- + +When $I_{D+1} = 0$, this equation is precisely the cobordism invariance of the partition function as implied by (1.11), since $[\partial W] = [\emptyset]$ by the definition of bordism groups. + +Now, by using $I_{D+1}$, we can define an element of $\operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R})$ by + +$$ (1.14) \qquad \Omega_{D+1}^S(X) \ni [W] \mapsto \int_W I_{D+1} \in \mathbb{R} $$ + +for any closed $(D+1)$-manifold $W$. Its well-definedness (i.e. it only depends on the equivalence class $[W]$ rather than a representative $W$) is immediate from the Stokes theorem and $dI_{D+1} = 0$. Moreover, for the partition function (1.13) to be well-defined, the integral $\int_W I_{D+1}$ on any closed $W$ must be an integer since $\partial W = \emptyset$ implies $Z(\partial W) = 1$. Therefore, (1.14) is actually an element of $\operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{Z})$, and, equivalently, it is in the kernel of the map $\operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R}) \to \operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R}/\mathbb{Z})$. For appropriate (but not all)$^4 S$, the fact that any element of $\operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{Z})$ can be realized in this way by some $I_{D+1}$ will follow from the Chern-Weil theory and the Hurewicz theorem as we will see later in the paper. This gives the exactness at $\operatorname{Hom}(\Omega_{D+1}^S(X), \mathbb{R})$. Also, if $I_{D+1} = dJ_D$ for some $J_D$ which is constructed by geometric data, we get $Z(\partial W) = \exp(2\pi\sqrt{-1}\int_{\partial W} J_D)$. This kind of contribution can be continuously deformed to zero by considering a one parameter family of QFT's parametrized by $t \in [0, 1]$ as + +$$ (1.15) \qquad Z_t(M) = \exp \left( 2\pi\sqrt{-1} \int_M t J_D \right), $$ + +so it does not contribute to the deformation classes of QFT's. This corresponds to taking the equivalence classes as in (1.8). The $\mathbb{R}/\mathbb{Z}$-valued functions $h$ which appeared in the definition of elements of $(I\Omega_{\mathrm{DR}}^G)^n(X)$ correspond to partition functions as $Z = \exp(2\pi\sqrt{-1}h)$, and $cw_g(f^*\alpha)$ in (1.7) corresponds to $J_D$. + +The authors are not aware of completely general proof of the expectation that the partition function can be expressed as (1.13). However, there are various evidence supporting this claim. First, (1.13) is exactly what is used in the construction of Wess-Zumino-Witten terms [WZ71] with the target space $X$ by extending a manifold $M$ to $W$ [Wit83]. In physics literature, Chern-Simons invariants are also described by (1.13). (See Example 4.81 for more precise discussions.) Second, invertible field theories constructed from massive fermions in the large mass limit satisfy (1.13). (See e.g. [WY19] for a systematic discussion). Third, other nontrivial examples of invertible QFT's also satisfy (1.13), such as the one relevant for the anomalies of chiral $p$-form fields [HTY20]. Finally, it is possible to give a physically reasonable derivation of a weaker version of the claim as follows. The functional derivative of the log of the partition function $\log Z(M)$ in terms of a background + +⁴The following statement fails when the Chern-Weil construction does not give an isomorphism. It happens in some noncompact groups, such as SL(2, R). Thus we have assumed that the groups $G_d$ in this paper are compact. However, there are also groups which are noncompact but the Chern-Weil isomorphism holds. An example is SL(2, Z) which has a trivial real cohomology $H^*(BSL(2, Z), R)$. This group can also have anomalies which are physically relevant [STY18, HTY19, HTY20]. +---PAGE_BREAK--- + +field $\phi$ (i.e. geometric data such as Riemann metric, connections, etc.) is +given by a one-point function of some local operator $O$, + +$$ +(1.16) \qquad \frac{\partial \log Z(M)}{\partial \phi(x)} = \langle O(x) \rangle, \quad (x \in M) +$$ + +In theories whose low energy limits are invertible QFT's, there is no light +degrees of freedom and all Feynman propagators are short range. Thus we +expect that the one-point function $\langle O(x) \rangle$ is given by local geometric data at +the point $x \in M$. Therefore, if two manifolds equipped geometric structure, +$M$ and $M'$, are homotopy equivalent, the ratio of their partition functions +is given by an integral of some local quantity.⁵ + +So far, we have argued that the exact sequence satisfied by $(I\Omega^S)^{D+1}(X)$ +is physically reasonable. However, the above physical arguments do not tell +us anything about the group $(I\Omega^S)^{D+1}(X)$ itself beyond the exact sequence. +The Anderson dual is defined in a very abstract way, and it is hard to find +a direct physical interpretation of the Anderson dual. One of our main +results stated in Theorem 1.9 is a natural isomorphism of the cohomology +theory $I\Omega^G$ to the theory $I\Omega_{\text{DR}}^G$. Here the structure type $\mathcal{S}$ is taken to be a +specific kind specified by $G$. The cohomology theory $I\Omega_{\text{DR}}^G$ is constructed in +a way which closely follows the above physical discussions. Therefore, our +results give a very strong support of the Freed-Hopkins conjecture 1.3 and +its further generalization to include target spaces $X$. + +Although $(I\Omega_{\mathrm{DR}}^{G})^{n}(X)$ is the group which is believed to classify the deformation classes of invertible field theories, the group $(\widetilde{I\Omega}_{\mathrm{DR}}^{G})^{n}(X)$ before taking the deformation classes is also physically very relevant. When we make background fields to dynamical fields, invertible field theories give topologically interesting terms in the action of the dynamical fields. Examples of this kind include topological $\theta$-terms in gauge theories and Wess-Zumino-Witten terms in sigma models. + +Finally, let us comment on reflection positivity. In the context of parti- +tion functions, reflection positivity is the following requirement. Consider a +*D*-manifold *M* with a boundary. We glue *M* and its opposite $\overline{M}$ along their +boundaries to get a closed manifold *M* ∪ *M* which is called a double. Reflec- +tion positivity is a requirement that *Z*(*M* ∪ *M*) is a nonnegative real number. +In the case of TQFT, reflection positivity is an important ingredient in the +classification of [FH21, Yon18]. Indeed, there are counterexamples to the +classification if we do not impose reflection positivity. See [FH21, HTY20] +for these examples. The partition function on a sphere *S**D* becomes neg- +ative, *Z*(*S**D*) = −1, although *S**D* can be constructed as the double of a +hemisphere. These examples have the property that *I**D*+1 discussed above +contains 1⁄2*E*, where *E* is the Euler density which gives the Euler number of +the manifold when it is integrated. The Euler density is excluded in this +paper by imposing a stability condition which we will define later in this +paper. But the stabilization is not important for the types of *G* considered +in [FH21]. We will make more comments on this point in Subsection 4.1.2. + +⁵This argument itself also applies to the case in which the theory is not invertible but is topologically ordered. Thus this argument does not give a complete proof of the claim. +---PAGE_BREAK--- + +1.3. **Mathematical significance.** In this subsection we explain our results from a mathematical point of view, especially its relation with the *differential cohomology theories*. One interesting point of the construction of $(\widehat{I\Omega}_{\mathrm{DR}}^G)^*$ lies in its similarity with the *differential character group* of Cheeger-Simons [CS85], which is a model for differential ordinary cohomology theory. + +Given a generalized cohomology theory $E^*$, its differential extension $\hat{E}^*$, +defined on manifolds, refines $E^*$ with additional differential-geometric data. +$\hat{E}^*$ itself is also called a *generalized differential cohomology theory*. For ex- +ample, $H^2(X; \mathbb{Z})$ classifies line bundles on $X$, whereas $\hat{H}^2(X; \mathbb{Z})$ classifies +hermitian line bundles with connections on $X$. See 2.5 for necessary back- +grounds. + +Theorem 1.9 says that the groups $(\widehat{I\Omega}_{\mathrm{DR}}^{G})^{*}$ is a model for a differential extension of $(I\Omega^{G})^{*}$. On the other hand, for the ordinary cohomology $HZ^{*}$, there are several known models for its differential extension, and the relevant one for us is the *differential character group* $\widehat{H}_{\mathrm{CS}}^{n}(-; \mathbb{Z})$ of Cheeger-Simons [CS85]. For a manifold $X$, $\widehat{H}_{\mathrm{CS}}^{n}(X; \mathbb{Z})$ consists of pairs $(\omega, k)$, where + +• $\omega \in \Omega_{\mathrm{clo}}^n(X)$, + +• $k: Z_{\infty,n-1}(X;\mathbb{Z}) \to \mathbb{R}/\mathbb{Z},$ + +• $\omega$ and $k$ satisfy the following compatibility condition. For any $c \in C_{\infty,n}(X;\mathbb{Z})$ we have + +$$ +k(\partial c) \equiv \langle c, \omega \rangle_X \pmod{\mathbb{Z}}. +$$ + +Here $Z_{\infty,*}$ and $C_{\infty,*}$ are the groups of smooth singular cycles and chains, re- +spectively. We immediately see that the definition of the group $(\widehat{I\Omega}_{\mathrm{DR}}^{G})^{n}(X)$ +explained in Subsection 1.1 is analogous to that of $\widehat{H}_{\mathrm{CS}}^{n}(X; \mathbb{Z})$. The essential +difference is the domain of $h$ or $k$, which, in the physical interpretation, plays +the role of the partition function of QFT's as explained in Subsection 1.2. +The authors feel that it is interesting that we found an analogy of differential +characters out of classifications of invertible QFT's. + +This is actually not just the analogy, but related to the *Anderson self-duality* of the ordinary cohomology. By the universal coefficient theorem we have $HZ^* \simeq IHZ^*$, with the *duality element* $\gamma \in IHZ^0(\mathrm{pt})$, equivalently, the natural transformation $\gamma: HZ^* \to (I\Omega^{\mathrm{fr}})* (= IZ^*)$. The above analogy allows us to construct the differential refinement $\hat{\gamma}_{\mathrm{DR}}^n : \hat{H}_{\mathrm{CS}}^*( - ; \mathbb{Z} ) \to (\hat{I}\Omega_{\mathrm{DR}}^{\mathrm{fr}})*(-)$ of the above transformation in Subsection 4.4. This turns out to be crucial when we analyze the elements in our model $(I\Omega_{\mathrm{DR}}^G)^*$ in Section 6. + +Finally we remark that in this paper we mainly focus on *tangential G-bordism theories*, as opposed to *normal G-bordism theories* which we denote by $\Omega^{G^\perp}$. But our results can be modified to give a model for the Anderson dual to normal *G*-bordism theories $(I\Omega^{G^\perp})^*$ in a straightforward way. This point is explained in Subsection 7.1. + +**1.4. The structure of the paper.** The paper is organized as follows. After we provide necessary preliminaries in Section 2, in Section 3 we introduce geometric stable tangential *G*-structures on manifolds. Using it, in Section 4 we define our models for $(I\Omega^G)^*$, and investigate into their properties. +---PAGE_BREAK--- + +Section 5 is devoted to the proof of our main theorem, which says that the constructions in Section 4 indeed give models for $(I\Omega^G)^*$. In Section 6 we investigate into examples coming from invertible QFT's. + +### 1.5. Notations. + +• By a topological space, we always mean a compactly generated topological space. + +• In this paper all manifolds are required to be *smooth*. + +• A *pair of topological spaces* means a pair $(X, A)$ with $X$ a topological space and $A$ its subspace. We denote $X := (X, \emptyset)$. We set + +$$ \text{Map}((X, A), (Y, B)) := \{f: X \to Y \mid f \text{ is continuous and } f(A) \subset B\}. $$ + +The category of pairs of topological spaces are denoted by TopPair. + +• A *pair of manifolds (X, A)* is a smooth manifold $X$ with a smooth submanifold $A$, which is a closed subset of $X$. We set + +$$ C^\infty((X, A), (Y, B)) := \{f: X \to Y \mid f \text{ is smooth and } f(A) \subset B\}. $$ + +The category of pairs of manifolds are denoted by MfdPair. + +• The category of CW-pairs are denoted by CWPair, and that of pointed CW complexes are denoted by CW\*. We regard CW\* as a subcategory of CWPair. + +• For two pairs of topological spaces $(X, A)$ and $(Y, B)$, their product is the pair of topological spaces given by + +$$ (X, A) \times (Y, B) := (X \times Y, X \times B \cup A \times Y) $$ + +Note that it restricts to the wedge product on pointed spaces, i.e., we have $(X, \{\text{pt}\}) \times (Y, \{\text{pt}\}) = (X, \{\text{pt}\}) \wedge (Y, \{\text{pt}\})$. + +• For a pair of manifolds $(X, A)$, we set + +$$ \Omega^n(X, A) := \{\omega \in \Omega^n(X; \mathbb{R}) \mid \omega|_A = 0\}. $$ + +• We also deal with differential forms with values in a graded real vector space $V^\bullet$. In the notation $\Omega^n(-; V^\bullet)$, $n$ means the total degree. In the case if $V^\bullet$ is infinite-dimensional, we topologize it as the colimit of all its finite-dimensional subspaces with the canonical topology, and set $\Omega^n(X; V^\bullet) := C^\infty(X; (\wedge T^*X \otimes_R V^\bullet)^n)$. This means that, any element in $\Omega^n(X; V^\bullet)$ can locally be written as a finite sum $\sum_i \xi_i \otimes \phi_i$ with $\xi_i \in \Omega^{m_i}(X)$ and $\phi_i \in V^{n-m_i}$ for some $m_i$ for each $i$. + +• For a real vector bundle $V$ with inner product over a topological space $X$, we denote by $D(V)$ and $S(V)$ the unit disk and sphere bundle of $V$, respectively. We denote $\tilde{D}(V) := D(V) \setminus S(V)$. The Thom space of $V$ is defined by + +$$ \mathrm{Thom}(V) := (D(V)/S(V), \{\mathrm{pt}\}) $$ + +which is regarded as a pair of topological spaces. We also use the smooth Thom space + +$$ (1.17) \qquad \mathrm{Thom}_\infty(V) := (D(V), S(V)), $$ + +which is a pair of manifolds if $X$ is smooth. +---PAGE_BREAK--- + +* For a manifold $X$ and a real vector space $V$, we denote by $\underline{V}$ the trivial bundle $\underline{V} := X \times V$ over $X$. + +* For a topological space $X$, we denote by $p_X: X \to \{\text{pt}\}$ the map to $\{\text{pt}\}$. + +* For two topological spaces $X$ and $Y$, we denote by $\text{pr}_X: X \times Y \to X$ the projection to $X$. + +* We set $D^1 = [-1, 1]$, $S^0 = \{-1, 1\}$ and $S^1 = [-1, 1]/\{-1, 1\}$. + +## 2. PRELIMINARIES + +**2.1. Spectra and generalized cohomology theories.** In this subsection we collect basic definitions and facts on spectra and their relations with generalized (co)homology theories. For details we refer to [Rud98]. + +A *spectrum* $X = (X_n, \sigma_n^X)_{n \in \mathbb{N}}$ consists of based topological spaces $X_n$ and based continuous maps $\sigma_n^X: \Sigma X_n \to X_{n+1}$, both indexed by $n \in \mathbb{N}$. We write $\tilde{\sigma}_n^X: X_n \to \Omega X_{n+1}$ for the based continuous map adjoint to $\sigma_n^X$. If $X = (X_n, \sigma_n^X)_{n \in \mathbb{N}}$ and $Y = (Y_n, \sigma_n^Y)$ are two spectra, a morphism of spectra $f = (f_n)_{n \in \mathbb{N}}: X \to Y$ is defined to be a sequence of based continuous maps $f_n: X_n \to Y_n$ ($n \in \mathbb{N}$) such that the following diagram commutes for every $n \in \mathbb{N}$: + +$$ +\begin{array}{ccc} +\Sigma X_n & \xrightarrow{\Sigma f_n} & \Sigma Y_n \\ +\bigg|_{\sigma_n^X} & & \bigg|_{\sigma_n^Y} \\ +X_{n+1} & \xrightarrow{f_{n+1}} & Y_{n+1} +\end{array} +$$ + +We write **Sp** for the category of spectra. + +Given a based topological space $X$, we define the *suspension spectrum* $\Sigma^\infty X$ by $(\Sigma^\infty X)_n = \Sigma^n X$ and $\sigma_n^{\Sigma^\infty X} = 1_{\Sigma^{n+1} X}: \Sigma(\Sigma^n X) \to \Sigma^{n+1} X$. This defines a functor $\Sigma^\infty: \text{Top}_* \to \text{Sp}$. We set $\mathcal{S} := \Sigma^\infty S^0$ and call it the *sphere spectrum*. + +We define the *suspension* $\Sigma X$ of a specrum $X$ by putting $(\Sigma X)_n = X_{n+1}$ and $\sigma_n^{\Sigma X} = \sigma_{n+1}^X$. This defines a functor $\Sigma: \text{Sp} \to \text{Sp}$. Note that $\Sigma(\Sigma^\infty X) = \Sigma^\infty(\Sigma X)$ for a based topological space $X$, which justifies the name ‘suspension’. + +We say that a spectrum $(X_n, \sigma_n^X)_{n \in \mathbb{N}}$ is an $\Omega$-spectrum if the adjoint structure map $\tilde{\sigma}_n^X: X_n \to \Omega X_{n+1}$ is a weak homotopy equivalence for every $n \in \mathbb{N}$. + +For $n \in \mathbb{Z}$, the *n*-th homotopy group of a spectrum $X$ is defined by + +$$ +\pi_n(X) = \varinjlim_p \pi_{n+p}(X_p) \left( \simeq \varinjlim_p \pi_{n+p}^{\mathrm{st}}(X_p) \right). +$$ + +We say that a morphism $f: X \to Y$ in **Sp** is a $\pi_*$-isomorphism if $f_*: \pi_n(X) \to \pi_n(Y)$ is an isomorphism for any $n \in \mathbb{Z}$. + +We define the *stable homotopy category of spectra* to be a localization of **Sp** at $\pi_*$-isomorphisms, i.e. a pair $(\text{Ho}(\text{Sp}), \gamma)$ satisfying the following properties: + +* $\text{Ho}(\text{Sp})$ is a category, and $\gamma: \text{Sp} \to \text{Ho}(\text{Sp})$ is a functor. + +* $\gamma$ sends the $\pi_*$-isomorphisms in $\text{Sp}$ to the isomorphisms in $\text{Ho}(\text{Sp})$. +---PAGE_BREAK--- + +• If $F: \mathbf{Sp} \to \mathcal{C}$ is a functor that sends the $\pi_*$-isomorphisms in $\mathbf{Sp}$ to the isomorphisms in $\mathcal{C}$, then there exists a unique functor $\bar{F}: \mathrm{Ho}(\mathbf{Sp}) \to \mathcal{C}$ such that $\bar{F} \circ \gamma = F$. + +Informally speaking, the category $\mathrm{Ho}(\mathbf{Sp})$ is a category obtained from $\mathbf{Sp}$ by 'formally inverting' $\pi_*$-isomorphisms. It is known that such a pair $(\mathrm{Ho}(\mathbf{Sp}), \gamma)$ indeed exists. One can easily see that $(\mathrm{Ho}(\mathbf{Sp}), \gamma)$ is unique up to unique isomorphism, and $\gamma$ defines a bijection between $\mathrm{Ob}(\mathrm{Ho}(\mathbf{Sp}))$ and $\mathrm{Ob}(\mathbf{Sp})$. We shall identify $\mathrm{Ob}(\mathrm{Ho}(\mathbf{Sp}))$ and $\mathrm{Ob}(\mathbf{Sp})$. Given two spectra $X$ and $Y$, we write $[X, Y]$ for the set of morphisms from $X$ to $Y$ in $\mathrm{Ho}(\mathbf{Sp})$. We shall always work in the category $\mathrm{Ho}(\mathbf{Sp})$, not in $\mathbf{Sp}$. + +We say that a sequence of morphisms in $\mathrm{Ho}(\mathbf{Sp})$ + +$$ X \xrightarrow{f} Y \xrightarrow{g} Z \xrightarrow{h} \Sigma X $$ + +is a *distinguished triangle* if it is isomorphic to some fiber sequence + +$$ \Sigma^{-1}B \to F \to E \to B. $$ + +The additive category $\mathrm{Ho}(\mathbf{Sp})$, equipped with the equivalence $\Sigma$: $\mathrm{Ho}(\mathbf{Sp}) \to \mathrm{Ho}(\mathbf{Sp})$ and the above class of distinguished triangles, is a *triangulated category*. In particular, it has the following properties. + +• Every morphism $f \in [X, Y]$ has a homotopy fiber $F(f)$, unique up to non-unique isomorphisms in $\mathrm{Ho}(\mathbf{Sp})$, which fits into a distinguished triangle + +(2.1) $$ F(f) \to X \xrightarrow{f} Y \to \Sigma F(f). $$ + +• Suppose that + +is a diagram in $\mathrm{Ho}(\mathbf{Sp})$ such that the two rows are distinguished and the left square commutes. Then, there exists a morphism $w: Z \to Z'$ in $\mathrm{Ho}(\mathbf{Sp})$ such that + +commutes. + +• For each distinguished triangle + +$$ X \xrightarrow{\tilde{f}} Y \xrightarrow{\tilde{g}} Z \xrightarrow{\tilde{h}} \Sigma X $$ + +and each object $W$ in $\mathrm{Ho}(\mathbf{Sp})$, the following sequence is exact. + +(2.2) $$ \dots \xrightarrow{-\Sigma^{-1}h_*} [W, X] \xrightarrow{f_*} [W, Y] \xrightarrow{g_*} [W, Z] \xrightarrow{h_*} [W, \Sigma X] \xrightarrow{-\Sigma f_*} \dots $$ + +The category $\mathrm{Ho}(\mathbf{Sp})$ also admits the structure of a *closed symmetric monoidal category*. Its unit object is the sphere spectrum $\mathbb{S}$, and the tensor product is called the *smash product* $\wedge$, which extends the smash product of +---PAGE_BREAK--- + +based spaces. For two objects $X, Y$ in $\text{Ho}(\mathbf{Sp})$, their internal hom is denoted by $F(X, Y)$ and called the *function spectrum*. We have the relation + +$$ (2.3) \qquad [X \wedge Y, Z] = [X, F(Y, Z)]. $$ + +An object $E$ in $\text{Ho}(\mathbf{Sp})$ defines a generalized cohomology theory $\{E^*, \delta^*\}$ and a generalized homology theory $\{E_*, \partial_*\}$ on the category of CW-pairs, and the corresponding reduced cohomology theory $\{\tilde{E}^*, s^*\}$ and reduced homology theory $\{\tilde{E}_*, s^*\}$ on the category of pointed CW complexes. Namely, for a pair of CW-complexes $(X, A)$ we set + +$$ E^n(X, A) = \tilde{E}^n(X/A) := [\Sigma^\infty(X/A), \Sigma^n E], \\ E_n(X, A) = \tilde{E}_n(X/A) := [\Sigma^n S, (\Sigma^\infty(X/A)) \wedge E], $$ + +and the structure maps are constructed in a canonical way. If $E$ is an $\Omega$-spectrum, we have $E^n(X, A) = \tilde{E}^n(X/A) = [X/A, E_n]$. + +Conversely, given a generalized cohomology theory $\{E^*, \delta^*\}$, by the *Brown representability theorem* there exists an $\Omega$-spectrum $E$ which represents $\{E^*, \delta^*\}$ in the above sense. + +Given a morphism $T \in [E, F]$ in $\text{Ho}(\mathbf{Sp})$, we also denote by $T: \{E^*, \delta^*\} \to \{F^*, \delta^*\}$ and $T: \{E_*, \partial_*\} \to \{F_*, \partial_*\}$ the induced natural transformations of generalized cohomology theories and homology theories. + +Suppose we have two $\Omega$-spectra $E$ and $F$. Natural transformations $T: \{E^*, \delta^*\} \to \{F^*, \delta^*\}$ between the corresponding generalized cohomology theories on CW-pairs are in one-to-one correspondence with the abelian group $\lim_{\leftarrow n} \tilde{F}^n(E_n)$. We have $\lim_{\leftarrow n} \tilde{F}^n(E_n) \neq [E, F]$ in general. Their difference is measured by the group $\lim_{\leftarrow n} \tilde{F}^{n-1}(E_n)$ by the following fact. + +**Fact 2.4** ([Rud98, Chapter III, Theorem 4.21]). Let $E$ and $F$ be two $\Omega$-spectra. Let $\{E^*, \delta^*\}$ and $\{F^*, \delta^*\}$ be the generalized cohomology theories on CW-pairs represented by $E$ and $F$, respectively. Denote by $\text{Hom}_{\text{coh}}(E^*, F^*)$ the abelian group of natural transformations of generalized cohomology theories from $\{E^*, \delta^*\}$ to $\{F^*, \delta^*\}$. Then we have the following exact sequence. + +$$ 0 \to \varinjlim_n \tilde{F}^{n-1}(E_n) \to [E, F] \to \text{Hom}_{\text{coh}}(E^*, F^*) \to 0. $$ + +**Fact 2.5.** [Ros94, p.135] For a sequence of abelian groups $A_* = (A_0 \to A_1 \to A_2 \to \dots)$ and an abelian group $B$, we have the following exact sequence. + +$$ 0 \to \varinjlim_n \text{Hom}(A_n, B) \to \text{Ext}^1(\varinjlim_n A_n, B) \to \varinjlim_n \text{Ext}^1(A_n, B) \to 0. $$ + +In particular if $B$ is an injective abelian group, we have $\varinjlim_n \text{Hom}(A_n, B) = 0$. + +**Remark 2.6.** A spectrum $E$ also defines a *generalized (co)homology theory on CW-spectra* by the formula $E^n(X) := [X, \Sigma^n E]$ and $E_n(X) := [\Sigma^n S, X \wedge E]$ for a CW-spectrum $X$. If we consider the set of natural transformations between two generalized cohomology theories on CW-spectra, the above difference vanishes. However, in this paper we always consider *generalized cohomology theories on CW-pairs*, unless otherwise stated. +---PAGE_BREAK--- + +For later use, we recall the explicit relation between the coboundary maps in an unreduced cohomology theory and the suspension isomorphisms in the corresponding reduced cohomology theory. For a generalized cohomology theory $\{E^*, \delta^*\}$ we have the suspension isomorphism + +$$ (2.7) \quad \text{susp}: E^n(X, A) \xrightarrow{\cong} E^{n+1}((S^1, \{\text{pt}\}) \times (X, A)) = \tilde{E}^{n+1}(\Sigma(X/A)) $$ + +for any CW-pair $(X, A)$. This map is given by the composition + +$$ (2.8) \quad E^n(X, A) \simeq E^n(\{1\} \times X \cup \text{cone}(A), \text{cone}(A)) \to E^n(\{1\} \times X \cup \text{cone}(A)) \\ \xrightarrow{\delta} E^{n+1}(\text{cone}(X), \{1\} \times X \cup \text{cone}(A)) \simeq E^{n+1}((S^1, \{\text{pt}\}) \times (X, A)). $$ + +Here we set $\text{cone}(Y) := ([{-1}, 1] \times Y)/(\{-1\} \times Y)$. The second map is given by the inclusion and the third map is the coboundary map. + +We would like to relate it with the cohomology of $S^1 \times (X, A)$. Let $k: \{\text{pt}\} \to S^1$ be the inclusion and $j: S^1 \to (S^1, \{\text{pt}\})$ be the map which comes from $\text{id}_{S^1}$. Then we have a split exact sequence + +$$ 0 \longrightarrow E^{n+1}((S^1, \{\text{pt}\}) \times (X, A)) \xrightarrow{(j \times \text{id})^*} E^{n+1}(S^1 \times (X, A)) \xrightarrow{(k \times \text{id})^*} E^{n+1}(\{\text{pt}\} \times (X, A)) \longrightarrow 0. $$ + +Thus we have a decomposition $E^{n+1}(S^1 \times (X, A)) = E^{n+1}((S^1, \{\text{pt}\}) \times (X, A)) \oplus E^{n+1}(\{\text{pt}\} \times (X, A))$ natural in $(X, A)$. So we have a homomorphism + +$$ (2.9) \quad \int : E^{n+1}(S^1 \times (X, A)) \xrightarrow{\text{pr}} E^{n+1}((S^1, \{\text{pt}\}) \times (X, A)) \xrightarrow{(\text{susp})^{-1}} E^n(X, A), $$ + +called the $S^1$-integration. + +**2.2. The Anderson duals.** In this subsection we collect basics on the Anderson duals for generalized cohomology theories. For more details, see for example [HS05, Appendix B], [FMS07, Appendix B]. In this subsection we entirely work with spectra. The corresponding statement for CW-pairs $(X, A)$ is obtained by considering the suspension spectrum $\Sigma^\infty(X/A)$. Remark that $\pi_*^{\text{st}}(X, A) = \pi_*(\Sigma^\infty(X/A))$. + +First note that the functor $X \mapsto \operatorname{Hom}(\pi_*(X), \mathbb{R}/\mathbb{Z})$ on $\operatorname{Ho(Sp)}^{\text{OP}}$ satisfies the Eilenberg-Steenrood axioms, so they are represented by an $\Omega$-spectrum denoted by $I(\mathbb{R}/\mathbb{Z})$. We also have the functor $X \mapsto \operatorname{Hom}(\pi_*(X), \mathbb{R})$, and the corresponding $\Omega$-spectrum $I\mathbb{R}$. By the Hurewicz isomorphism we have $\operatorname{Hom}(\pi_*(X), \mathbb{R}) = H^*(X; \mathbb{R})$ and $I\mathbb{R}$ is isomorphic to the Eilenberg-MacLane spectrum $H\mathbb{R}$. Therefore we just take $I\mathbb{R} = H\mathbb{R}$. The morphism in $\operatorname{Ho(Sp)}$ representing the transformation $\operatorname{Hom}(\pi_*(-), \mathbb{R}) \to \operatorname{Hom}(\pi_*(-), \mathbb{R}/\mathbb{Z})$ is denoted by + +$$ (2.10) \qquad \pi: H\mathbb{R} \to I(\mathbb{R}/\mathbb{Z}). $$ + +**Definition 2.11** (The Anderson dual to the sphere spectrum). The Anderson dual to the sphere spectrum, $IZ$, is the homotopy fiber (see (2.1)) of the map (2.10). +---PAGE_BREAK--- + +This means that, we have a spectrum $HZ$ together with two morphisms +$p: \Sigma^{-1}I(\mathbb{R}/\mathbb{Z}) \to I\mathbb{Z}$ and $\iota: I\mathbb{Z} \to H\mathbb{R}$ in $Ho(\mathbf{Sp})$ such that + +$$ +(2.12) \qquad \Sigma^{-1}I(\mathbb{R}/\mathbb{Z}) \xrightarrow{p} I\mathbb{Z} \xrightarrow{\iota} H\mathbb{R} \xrightarrow{\pi} I(\mathbb{R}/\mathbb{Z}) +$$ + +is a distinguished triangle. Applying the exact sequence for a distinguished +triangle in (2.2) in this case, for each spectrum X we get the following exact +sequence. + +$$ +(2.13) \qquad \dots \to H^{n-1}(X; \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\pi_{n-1}(X), \mathbb{R}/\mathbb{Z}) \xrightarrow{p} I\mathbb{Z}^n(X) \\ +\qquad \to H^n(X; \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\pi_n(X), \mathbb{R}/\mathbb{Z}) \to \dots \text{ (exact)}. +$$ + +**Definition 2.14** (The Anderson dual to a spectrum). Let $E$ be a spectrum. +The *Anderson dual to* $E$, denoted by $IE$, is a spectrum defined as the +function spectrum from $E$ to $IZ$, + +$$ +IE := F(E, I\mathbb{Z}). +$$ + +By (2.3), the exact sequence (2.13) implies that the following sequence is +exact. + +$$ +(2.15) \qquad \dots \to \operatorname{Hom}(E_{n-1}(X), \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(E_{n-1}(X), \mathbb{R}/\mathbb{Z}) \xrightarrow{p} IE^n(X) \\ +\qquad \to \operatorname{Hom}(E_n(X), \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(E_n(X), \mathbb{R}/\mathbb{Z}) \to \dots (\text{exact}). +$$ + +If $E$ is a ring spectrum, the resulting cohomology theory $E^*$ is multiplicative. +The cohomology theories in the exact sequence (2.15) are module theories +over $E^*$, and the sequence is compatible with the module structure. + +Note that, in the case $E = HZ$, replacing $IHZ$ by $HZ$ in the sequence (2.15), we also get the exact sequence by the universal coefficient theorem. +Actually, this phenomenon is called the *Anderson self-duality of HZ*, $IHZ \simeq HZ$. + +We take the morphism Hur ∈ [S, HZ] which gives the Hurwicz homomor- +phism π∗(X) → H∗(X). The morphism Hur and the distinguished triangle +(2.12) give a commutative diagram + +$$ +(2.16) \qquad \begin{tikzcd}[column sep=2.8em, row sep=2.8em] +\Sigma^{-1}F(HZ, \mathbb{R}) \arrow[r, "-(\Sigma^{-1}\pi)_*"] & \Sigma^{-1}F(HZ, I(\mathbb{R}/\mathbb{Z})) \arrow[r, "p_*"] & F(HZ, I\mathbb{Z}) \arrow[r, "l_*"] & F(HZ, \mathbb{R}) \\ +\bigdownarrow_{Hur^*} & \bigdownarrow_{Hur^*} & \bigdownarrow_{Hur^*} & \bigdownarrow_{Hur^*} +\\ +\Sigma^{-1}F(S, \mathbb{R}) \arrow[r, "-(\Sigma^{-1}\pi)_*"] & \Sigma^{-1}F(S, I(\mathbb{R}/\mathbb{Z})) \arrow[r, "p_*"] & F(S, I\mathbb{Z}) \arrow[r, "l_*"] & F(S, \mathbb{R}) +\end{tikzcd} +$$ + +Notice that $F(HZ, \mathbb{R}) \simeq H\mathbb{R}$, $F(HZ, I(\mathbb{R}/\mathbb{Z})) \simeq H(\mathbb{R}/\mathbb{Z})$ and $F(HZ, I\mathbb{Z}) \simeq H\mathbb{Z}$ where the last isomorphism is the Anderson self-duality. Under this isomorphism, the top row of (2.16) is the exact triangle for the extension of coefficient groups for the ordinary cohomology. Also, for any spectrum $X$ we have $F(S, X) \simeq X$ since $S$ is the unit of $Ho(\mathbf{Sp})$. Moreover, recall the fact that $\mathbb{R} = H\mathbb{R}$ by the Hurewicz isomorphism. + +**Definition 2.17.** Using the isomorphisms of spectra explained above, we define $\gamma_{\mathbb{R}/\mathbb{Z}} \in [\Sigma^{-1}H(\mathbb{R}/\mathbb{Z}), \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})]$ and $\gamma \in [HZ, I\mathbb{Z}]$ to be the morphism corresponding to the second and the third vertical arrows in (2.16), +---PAGE_BREAK--- + +respectively. The element $\gamma$ is called the *duality element* for $HZ$. Also define $p_H \in [\Sigma^{-1}H(\mathbb{R}/\mathbb{Z}), HZ]$ and $\iota_H \in [H\mathbb{R}, IZ]$ to be the morphisms corresponding to the middle and the right horizontal arrows in the top row in (2.16). + +By the commutativity of (2.16), we get + +**Lemma 2.18.** The following diagram in $\operatorname{Ho}(Sp)_{\mathbb{R}}$ commutes. + +$$ +\begin{tikzcd} +\Sigma^{-1}H\mathbb{R} \arrow[r] & \Sigma^{-1}H(\mathbb{R}/\mathbb{Z}) \arrow[r, "$p_H$"] & HZ \arrow[r, "$\iota_H$"] & H\mathbb{R}, \\ +\Sigma^{-1}I\mathbb{R} \arrow[r, "$-\Sigma^{-1}\pi$"] & \Sigma^{-1}I(\mathbb{R}/\mathbb{Z}) \arrow[r, "$p$"] & IZ \arrow[r, "$\iota$"] & I\mathbb{R} +\arrow[onnee={,yshift=-2.5ex},{xshift=1.5cm}] +\arrow@{}[\wherefore] +\arrow@{}[left, "$\gamma_{\mathbb{R}/\mathbb{Z}}$"'] +\arrow@{}[right, "$\gamma$"'] +\arrow@{}[above, "$\iota_H$"'] +\arrow@{}[below, "$\iota$"'] +\arrow@{}[left, "$\gamma_{\mathbb{R}/\mathbb{Z}}$"'] +\arrow@{}[right, "$\gamma$"'] +\arrow@{}[above, "$\iota_H$"'] +\arrow@{}[below, "$\iota$"'] +\arrow[dashed, "$\sim$"'] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$"] +\arrow[dashed, "$\sim$] +\arrow[dashed, "$\sim$] +\arrow[dashed, "$\sim$] +\arrow[dashed, "$\sim$] +$$ + +The duality element $\gamma$ makes the following diagram commute. + +(2.19) + +$\iota_* : [HZ, IZ] \to [HZ, H\mathbb{R}]$ + +is injective. In particular, the commutativity of (2.19) uniquely characterizes $\gamma$. + +*Proof.* By the exact sequence for the distinguished triangle (2.12), the following sequence is exact. + +$$ [HZ, \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})] \xrightarrow{p_*} [HZ, IZ] \xrightarrow{\iota_*} [HZ, H\mathbb{R}]. $$ + +Since we have $[HZ, \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})] = \operatorname{Hom}(\pi_1(HZ), \mathbb{R}/\mathbb{Z}) = 0$, we get the desired injectivity. $\square$ + +**2.3. Madsen-Tillmann spectra and stable tangential G-bordism theories.** In this subsection we explain the Madsen-Tillmann spectra and their relation with the stable tangential *G*-bordism theories, following [Fre19, Section 6.6]. + +Let $G = \{G_d, s_d, \rho_d\}_{d \in \mathbb{Z}_{>0}}$ be a sequence of compact Lie groups equipped with homomorphisms $s_d: G_d \to G_{d+1}$ and $\rho_d: G_d \to O(d, \mathbb{R})$ for each $d$ such that the following diagram commutes. + +$$ +\xymatrix{ +G_d \ar[r, "\rho_d"] & O(d, \mathbb{R}) . \ar[l] \\[1ex] +s_d \ar[r, "ccirc", "?"'] & \\[-1.5ex] +G_{d+1} \ar[r, "\rho_{d+1}"'] & O(d+1, \mathbb{R}) +} +$$ + +Here we use the inclusion $O(d, \mathbb{R}) \hookrightarrow O(d+1, \mathbb{R})$ defined by + +$$ A \mapsto \left[ \begin{array}{c|c} 1 & 0 \\ \hline 0 & A \end{array} \right] $$ + +throughout this paper. Examples of such $G$ are given in Example 2.30 below. +---PAGE_BREAK--- + +For two nonnegative integers $d$ and $D$ satisfying $d \le D$, we denote by $\text{Gr}_d(D)$ the $d$-dimensional Grassmannian of $\mathbb{R}^D$, which is the space of $d$-dimensional subspace of $\mathbb{R}^D$. We use the model of classifying space of $\text{BO}(d, \mathbb{R})$ constructed as + +$$ \text{BO}(d, \mathbb{R}) = \varinjlim_q \text{Gr}_d(d+q). $$ + +On $\text{Gr}_d(d+q)$, we have the tautological bundle $S_d$, which is a rank-$d$ sub-bundle of the trivial rank-$d+q$ vector bundle $\mathbb{R}^{d+q}$. We define $Q_q$ to be the rank-$q$ quotient bundle $\mathbb{R}^{d+q}/S_d$ over $\text{Gr}_d(d+q)$. + +Take a universal $G_d$ bundle $EG_d \to BG_d$ for each $d$ and take classifying maps $B\rho_d: BG_d \to \text{BO}(d, \mathbb{R})$ and $Bs_d: BG_d \to BG_{d+1}$ such that the following diagram commutes. + +Here the right vertical arrow is induced by $\text{Gr}_d(d+q) \hookrightarrow \text{Gr}_{d+1}(d+1+q)$. We take $B\rho_d$ to be a Serre fibration, which is always possible (we will use this assumption in Lemma 2.24). For each $q$ we define the space $\mathcal{X}_d(d+q)$ as the pullback, + +$$ (2.21) \qquad \begin{tikzcd} \mathcal{X}_d(d+q) \arrow[r] & \text{Gr}_d(d+q) \\[1ex] BG_d \arrow[u] \arrow[r, "B\rho_d"] & \text{BO}(d, \mathbb{R}) \arrow[u] \\[1ex] \mathcal{X}_d(d+q) \arrow[u] \arrow[r, "\alpha_{q,d}"'] & \mathcal{X}_d(d+q+1) \end{tikzcd} . $$ + +We also denote the pullback of the bundle $Q_q$ to $\mathcal{X}_d(d+q)$ by $Q_q \to \mathcal{X}_d(d+q)$. We have the maps $\alpha_{q,d}: \mathcal{X}_d(d+q) \to \mathcal{X}_d(d+q+1)$ and $\beta_{q,d}: \mathcal{X}_d(d+q) \to \mathcal{X}_{d+1}(d+1+q)$, induced by $\text{Gr}_d(d+q) \hookrightarrow \text{Gr}_d(d+q+1)$ and $\text{Gr}_d(d+q) \hookrightarrow \text{Gr}_{d+1}(d+1+q)$ respectively. The pullback bundle $\alpha_{q,d}^*Q_{q+1}$ is canonically identified with $Q_q \oplus \mathbb{R}$, + +Since we have + +$$ \Sigma\text{Thom}(Q_q \to \mathcal{X}_d(d+q)) \simeq \text{Thom}(Q_q \oplus \mathbb{R} \to \mathcal{X}_d(d+q)), $$ + +we get a map of Thom spaces + +$$ (2.22) \qquad \Sigma\text{Thom}(Q_q \to \mathcal{X}_d(d+q)) \to \text{Thom}(Q_{q+1} \to \mathcal{X}_d(d+q+1)). $$ + +**Definition 2.23** (Madsen-Tillmann spectrum). The Madsen-Tillmann spectrum MTG$_d$ is a spectrum whose $(d+q)$-th space is given by + +$$ (MTG_d)_{d+q} := \text{Thom}(Q_q \to \mathcal{X}_d(d+q)) $$ + +and structure maps are given by (2.22). For $n < d$, we set $(MTG_d)_n$ to be the one-point space. +---PAGE_BREAK--- + +For later use, we note the following. + +**Lemma 2.24.** The map $\mathcal{X}_d(d+q) \to BG_d$ is $q$-connected. + +*Proof.* Since the Stiefel manifold $V_d(\mathbb{R}^{d+q})$ is $(q-1)$-connected, the map $\text{Gr}_d(d+q) \to \text{BO}(d, \mathbb{R})$ is $q$-connected. We have assumed that $B\rho_d: BG_d \to \text{BO}(d, \mathbb{R})$ is a Serre fibration, so by (2.21) we see that the map $\mathcal{X}_d(d+q) \to BG_d$ is also $q$-connected. $\square$ + +Passing to the Thom spaces (see [Rud98, Chapter IV, Lemma 5.20]), + +Lemma (2.24) implies that, + +**Lemma 2.25.** When $q \ge 2$, the map $\Sigma^\infty(MTG_d)_{d+q} \to \Sigma^{d+q}MTG_d$ is $2q$-connected. + +Next we stabilize the above construction with respect to $d$. We have the +pullback diagram, + +This gives the map $\Sigma^d MTG_d \to \Sigma^{d+1} MTG_{d+1}$ of spectra. + +**Definition 2.27** (Stable Madsen-Tillmann spectrum). For $G = \{G_d, s_d, \rho_d\}_{d \in \mathbb{N}}$ +as above, the *stable Madsen-Tillmann spectrum* *MTG* is defined by + +$$ +\mathrm{MTG} := \operatorname{holim}_d \Sigma^d \mathrm{MTG}_d. +$$ + +By Lemma 2.25, we have + +**Lemma 2.28.** For $q \ge 2$, the map $\Sigma^\infty(MTG)_q \to \Sigma^q MTG$ is $2q$-connected. + +The generalized homology theory represented by *MTG* is the *stable tangential G-bordism theory* $\Omega_*^G$. Here we recall the definition of the *stable tangential n-dimensional G-bordism group* $\Omega_n^G(X, A)$ for a pair of topological spaces $(X, A)$. A *stable tangential G-cycle* of dimension $n$ over $(X, A)$ is a triple $(M, \tilde{g}, f)$, where + +• $M$ is an $n$-dimensional compact manifold with boundary, + +• $\tilde{g} = (d, P, \psi)$ is a *stable tangential G-structure* on $M$, i.e., $d \ge n$ is an integer, $P$ is a principal $G_d$-bundle $M$ and $\psi: P \times_{\rho_d} \mathbb{R}^d \simeq \mathbb{R}^{d-n} \oplus TM$ is an isomorphism of vector bundles between the associated bundle to $P$ and the stable tangent bundle of rank $d$. + +• $f \in \operatorname{Map}((M, \partial M), (X, A))$. + +Actually, we are going to work with *geometric stable tangential G-cycles* in +this paper. The definition and various constructions are given in Section +3 below. A geometric stable tangential *G-cycle* is given by just adding a +data of *Gd-connection* $\nabla$ on the principal *Gd-bundle* $P$ to a stable tangen- +tial *G*-cycle. In particular, the definition of the *stabilization* and the *bordism +relations* for stable tangential *G*-cycles are given by just forgetting the infor- +mation on connections in the corresponding definition for geometric stable +tangential *G*-cycles (Definition 3.6 and Definition 3.8), so we omit the details +here. +---PAGE_BREAK--- + +We introduce an equivalence relation $\sim_{\text{bord}}$ on stable tangential $G$-cycles generated by the following relations. + +• Isomorphisms. + +• $(M, \tilde{g}, f) \sim_{\text{bord}} (M, \tilde{g}(1), f)$. Here $\tilde{g}(1)$ is the stabilization of $\tilde{g}$. + +• $(M, \tilde{g}, f) \sim_{\text{bord}} \emptyset$ if $(M, \tilde{g}, f)$ is *null-bordant*, meaning that there exists a stable tangential $G$-cycle $(W, g_W, f_W)$ of dimension $n$ over $(X, X)$ which is bounded by $(M, \tilde{g}, f)$. + +**Definition 2.29** (The stable tangential $G$-bordism group). The stable tangential $G$-bordism group $\Omega_n^G(X, A)$ is defined as the abelian group consisting of the equivalence classes under $\sim_{\text{bord}}$ of stable tangential $G$-cycles of dimension $n$ over $(X, A)$. Here the abelian group structure is given by the disjoint union. + +*Example 2.30.* Here are some examples of $G$ and the corresponding stable tangential $G$-structures. + +(1) The case where $G_d = \{1\}$ for all $d$. Since the stable tangential $G$-structure is a stable tangential framing in this case, we denote this $G$ by fr. + +(2) $\text{SO} := \{\text{SO}(d, \mathbb{R})\}_d$ with $\rho_d$ and $s_d$ given by the inclusions. A stable tangential SO-structure is an orientation with a Riemannian metric. + +(3) $\text{Spin} := \{\text{Spin}(d)\}_d$ with $\rho_d: \text{Spin}(d) \to \text{O}(d, \mathbb{R})$ given by the double covering of $\text{SO}(d, \mathbb{R})$ composed with the inclusion, and $s_d$ given by the inclusion. A stable tangential Spin-structure is a Spin-structure in the usual sense. + +(4) Let $H$ be a compact Lie group, and set $\text{SO} \times H := \{\text{SO}(d, \mathbb{R}) \times H\}_d$ with $\rho_d$ given by the composition of the projection $\text{SO}(d, \mathbb{R}) \times H \to \text{SO}(d, \mathbb{R})$ and the inclusion, and $s_d$ given by the inclusion. A stable tangential $SO \times H$-structure is an orientation with a Riemannian metric, together with a choice of principal $H$-bundle. This group $H$ is called the *internal symmetry group* in physics. + +It is known that $\Omega_*^G$ is a generalized homology theory, called the *stable tangential $G$-bordism theory*. We also just call it the *$G$-bordism theory* in this paper, assuming that “stable tangential” is understood. In fact, we have the following fact. + +**Fact 2.31.** The stable tangential $G$-bordism theory $\Omega_*^G$ is represented by the stable Madsen-Tillmann spectrum MTG. + +This means that, for an integer $n$ and $(X, x)$ a pointed topological space, +we have + +$$ +\pi_n(X \wedge MTG) \simeq \Omega_n^G(X). +$$ + +The proof of Fact 2.31 is given by constructing the homomorphism $\pi_n(X \wedge MTG) \simeq \Omega_n^G(X)$ by the Pontryagin-Thom construction explained in [Fre19, Remark 6.65]$^6$. We explain the geometric Pontryagin-Thom construction + +⁶Precisely speaking, [Fre19, Remark 6.65] explains the construction in the case of the unstable Madsen-Tillmann spectrum *MTG**d*. However it is easily modified to work in our setting, using the fact that πn(*MTG*) = lim→d πn(*Σ**d**MTG**d*). +---PAGE_BREAK--- + +which refines the usual Pontryagin-Thom construction with the data of connections in Subsection 3.2. + +In this paper we are interested in the Anderson dual to the stable tangential $G$-bordism theory $\{(I\Omega^G)^*, \delta^*\}$. Definition 2.14 applied to $E = MTG$, it is represented by the function spectrum + +$$ (2.32) \qquad I\Omega^G = F(MTG, IZ). $$ + +By (2.3), for a spectrum X we have + +$$ (I\Omega^G)^n(X) = [X \wedge MTG, \Sigma^n IZ]. $$ + +The long exact sequence for the Anderson dual theory (2.15) becomes + +$$ \begin{align*} +(2.33) \quad & \cdots \to \operatorname{Hom}(\Omega_{n-1}^G(X), \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\Omega_{n-1}^G(X), \mathbb{R}/\mathbb{Z}) \xrightarrow{p} (I\Omega^G)^n(X) \\ +& \to \operatorname{Hom}(\Omega_n^G(X), \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\Omega_n^G(X), \mathbb{R}/\mathbb{Z}) \to \cdots (\text{exact}). +\end{align*} $$ + +**2.4. Singular (co)homology.** In this subsection we set up our conventions on the singular homology and cohomology, in particular those with local coefficients. We refer to [Hat01] for details. + +A local coefficient system $E$ over a topological space $X$ means a flat bundle of abelian groups over $X$. Given such $E$, a singular $n$-chain with coefficient in $E$ is a finite sum of $\tilde{\sigma}_i$'s, where each $\sigma_i: \Delta^n \to X$ is a singular $n$-simplex in $X$ and $\tilde{\sigma}_i: \Delta^n \to E$ is a parallel lift of $\sigma_i$. These chains form an abelian group denoted by $C_n(X; E)$. For a pair of topological spaces $(X, A)$, we denote by $C_n(X, A; E) := C_n(X; E)/C_n(A; E)$, $\partial: C_n(X, A; E) \to C_{n-1}(X, A; E)$, $Z_n(X, A; E)$ and $H_*^{sing}(X, A; E)$ the relative chains, boundary map, cycles and homology, respectively. + +For cohomology, we define $C^n(X; E)$ to be the group consisting of functions $\phi$ assigning to each singular simplex $\sigma: \Delta^n \to X$ a parallel lift $\phi(\sigma): \Delta^n \to E$. An element of this group is called a singular $n$-cochain with coefficient in $E$. We denote by $C^n(X, A; E) := \ker(C^n(X; E) \to C^n(A; E))$, $\delta: C^n(X, A; E) \to C^{n+1}(X, A; E)$, $Z^n(X, A; E)$, and $H_*^{sing}(X, A; E)$ the relative cochains, coboundary map, cocycles and cohomology, respectively. + +The cup and cap products we use in general has the following form. Let $N$ be an abelian group and $E$ and $F$ be two local systems on $X$ of $N$-modules. Then we have cup and cap products, + +$$ \begin{align*} +\cup: C^k(X, A; E) \otimes_N C^l(X, A; F) &\to C^{k+l}(X, A; E \otimes_N F), \\ +\cap: C^l(X; F^*) \otimes_N C_k(X; E \otimes_N F) &\to C_{k-l}(X; E). +\end{align*} $$ + +Here we use the convention on the cap product so that $\phi_1 \cap (\phi_2 \cap \sigma) = (\phi_1 \cup \phi_2) \cap \sigma$ holds, namely in the case of the trivial coefficient, given by + +$$ \phi \cap \sigma = \phi(\sigma|_{[v_{k-l}, \dots, v_k]})\sigma|_{[v_0, \dots, v_{k-l}]}. $$ + +In this paper we deal with the following examples of local systems. + +*Example 2.34 (Orientation bundles of vector bundles)*. Let $X$ be a manifold and $\pi: V \to X$ be a real vector bundle over $X$. We define + +$$ \operatorname{Ori}_{\mathbb{Z}}(\pi) := \operatorname{Fr}(\pi) \times_{\operatorname{sign}} \mathbb{Z}, \quad \operatorname{Ori}(\pi) := \operatorname{Ori}_{\mathbb{Z}}(\pi) \otimes \mathbb{R}. $$ +---PAGE_BREAK--- + +Here $\mathrm{Fr}(\pi)$ is the frame bundle of $\pi: V \to X$. This gives a local system over $X$. For a manifold $X$, we denote $\mathrm{Ori}_Z(X) := \mathrm{Ori}_Z(TX \to X)$ and $\mathrm{Ori}(X) := \mathrm{Ori}(TX \to X)$. Note that in our convention $\mathrm{Ori}(\pi)$ is a bundle of $\mathbb{R}$-modules. By abuse of notation we also denote $\mathrm{Ori}(\pi) := \pi^*\mathrm{Ori}(\pi)$, which is a local coefficient system on $V$. + +If we have two real vector bundles $\pi_1$ and $\pi_2$ over $X$, when their direct sum is written in the order $\pi_1 \oplus \pi_2$, we use the identification + +$$ (2.35) \qquad \mathrm{Ori}(\pi_1) \otimes \mathrm{Ori}(\pi_2) \simeq \mathrm{Ori}(\pi_1 \oplus \pi_2) $$ + +induced by the homomorphism of the frame bundles, + +$$ (2.36) \qquad \mathrm{Fr}(\pi_1) \times_X \mathrm{Fr}(\pi_2) \to \mathrm{Fr}(\pi_1 \oplus \pi_2) $$ + +$$ (2.37) \qquad ( \{e_1, \dots, e_i \}, \{f_1, \dots, f_j\} ) \mapsto \{e_1, \dots, e_i, f_1, \dots, f_j\}. $$ + +We have $\mathrm{Ori}(\pi)^* \simeq \mathrm{Ori}(\pi)$ canonically. Thus as an example of cap product we get the pairing, + +$$ \langle , \rangle_X : C^k(X; \mathrm{Ori}(X)) \otimes C_k(X; \mathrm{Ori}(X)) \to \mathbb{R}. $$ + +We also denote it just by $\langle , \rangle$ when $X$ is understood from the context. + +*Example 2.38 (The fundamental classes of manifolds)*. For an $n$-dimensional compact manifold $X$ with boundary, we have the *fundamental class* $[X] \in H_n(X, \partial X; \mathrm{Ori}_Z(X))$. We also denote its image under tensoring $\mathbb{R}$ by $[X] \in H_n(X, \partial X; \mathrm{Ori}(X))$ and call it the fundamental class. + +For a manifold with boundary $(X, \partial X)$, we identify $\mathrm{Ori}(X)|_{\partial X} \simeq \mathrm{Ori}(\partial X)$ as follows. Choose a collar structure $U \simeq (-\epsilon, 0] \times \partial X$ with $U$ an open neighborhood of $\partial X$ in $X$ (see Subsection 2.6 for more on this). This gives the identification $TX|_{\partial X} \simeq \mathbb{R} \oplus T\partial X$. Since we have $\mathrm{Ori}(\mathbb{R}) \simeq \mathbb{R}$ canonically, the composition + +$$ (2.39) \qquad \mathrm{Ori}(X)|_{\partial X} \simeq \mathrm{Ori}(\mathbb{R} \oplus T\partial X) \simeq \mathrm{Ori}(\mathbb{R}) \otimes \mathrm{Ori}(\partial X) \simeq \mathrm{Ori}(\partial X) $$ + +gives the desired isomorphism. Under this identification, we have $\partial[X] = [\partial X]$. + +*Example 2.40 (The Thom classes of real vector bundles)*. Let $\pi: V \to X$ be a real vector bundle of rank $r$ over a topological space $X$. Choosing any inner product on $V$, we have the *Thom class* $\tau_\pi \in H^r(\mathrm{Thom}(V); \mathrm{Ori}_Z(\pi))$. This element gives the Thom isomorphism $H^{*+r}(\mathrm{Thom}(V); \mathrm{Ori}_Z(\pi)) \simeq H^*(X; \mathbb{Z})$. We also denote the image of $\tau_\pi$ under tensoring $\mathbb{R}$ by $\tau_\pi \in H^r(\mathrm{Thom}(V); \mathrm{Ori}(\pi))$ and call it the Thom class. + +Now we turn to the smooth singular homology and cohomology. From now on we assume that $(X, A)$ is a pair of manifolds. By requiring singular simplices $\sigma: \Delta^n \to X$ to be smooth in the constructions above, we get smooth relative singular chain and cochain complexes $\{C_{\infty,*}(X, A; E), \partial\}$ and $\{C^*_{\infty}(X, A; E), \delta\}$, and get relative smooth singular homology and cohomology groups $H_{*,\infty}^{\text{sing}}(X, A; E)$ and $H_{*,\infty}^*(X, A; E)$, respectively. The groups of relative cycles and cocycles are denoted by $Z_{\infty,*}(X, A; E)$ and $Z^*_{\infty}(X, A; E)$, respectively. We have the inclusion + +$$ (2.41) \qquad C_{\infty,n}(X, A; E) \subset C_n(X, A; E), $$ +---PAGE_BREAK--- + +and the restriction map + +$$ +(2.42) \qquad C^n(X, A; E) \to C_\infty^n(X, A; E). +$$ + +In the case where $E$ is a local system of $\mathbb{R}$-modules we also have the inclusion, + +$$ +(2.43) \qquad \Omega^n(X, A; E) \subset C_\infty^n(X, A; E) +$$ + +In this paper we use the transformations (2.41), (2.42) and (2.43) without reference, i.e., given $\omega \in \Omega^n(X, A; E)$, we also denote by $\omega \in C^\infty_\infty(X, A; E)$ the corresponding element, and so on. These maps are compatible with the boundary and coboundary maps, and induce the corresponding maps on homology and cohomology groups. These maps gives the isomorphisms on the (co)homology levels, $H_n^{\text{sing}} \simeq H_n^{\text{sing},\infty}$ and $H_{\text{sing}}^n \simeq H_{\text{sing},\infty}^n \simeq H_{\text{dR}}^n$. + +However, we need to take some care when dealing with product structures. +Namely, the inclusion (2.43) does not map the wedge products to the cup +products, $\omega \wedge \eta \neq \omega \cup \eta$, although they coincide on the cohomology level. +For our purposes, we want a natural cochain homotopy between $\wedge$ and $\cup$. +Such a homotopy exists by [Gug76]. Explicitly, for each pair of nonnegative +integers $r$ and $s$, there exists a natural transformation$^7$ on MfdPair$^{\text{op}}$, + +$$ +(2.44) \qquad B : \Omega^r(X, A) \otimes \Omega^s(X, A) \to C_\infty^{r+s-1}(X, A) +$$ + +with, for any $\omega \in \Omega^r(X, A)$ and $\eta \in \Omega^s(X, A)$, + +$$ +(2.45) \qquad \delta B(\omega, \eta) + B(d\omega, \eta) + (-1)^r B(\omega, d\eta) = \omega \wedge \eta - \omega \cup \eta. +$$ + +Any two choices of such $B$ are naturally cochain homotopic. Such a ho- +motopy is also used in [CS85, (1.10)] and [HS05, (3.8)], for example. One +way to realize it is to use the barycentric subdivision on simplexes to make +a homotopy between $\wedge$ and the $\cup$ composed with the shuffle map (see the +construction of [CS85, (1.10)], but note that they use the normalized cubic +chains), and then use a homotopy to the cup product. By this explicit con- +struction, the map (2.44) can be extended to the case with local coefficient +systems, + +$$ +(2.46) \qquad B: \Omega^r(X, A; E) \otimes \Omega^s(X, A; F) \to C_\infty^{r+s-1}(X, A; E \otimes_N F) +$$ + +with the same property (2.45), satisfying the naturality with respect to the +pullback of coefficient systems. + +In some parts of the paper, we need a natural way to produce a singular +$n$-chain on $I \times X$ out of $(n-1)$-chain on $X$ for a closed interval $I = [a, b]$ in +$\mathbb{R}$. This is given by the *Prism operator* in [Hat01, Proof of Theorem 2.10]. +Let $E$ be a local coefficient system on $X$. Then the Prism operator + +$$ +(2.47) \qquad P_I : C_{n-1}(X; E) \to C_n(I \times X; \mathrm{pr}_X^* E) +$$ + +is defined as follows. Let $\sigma: \Delta^{n-1} \to X$ be a simplex and $\tilde{\sigma}: \Delta^{n-1} \to E$ +is its parallel lift. The product $I \times \Delta^{n-1}$ contains $2n$-vertices. We denote +the vertices on $\{a\} \times \Delta^{n-1}$ as $v_0, \dots, v_{n-1}$, and those on $\{b\} \times \Delta^{n-1}$ as + +⁷The result in [Gug76] applies in the absolute case $A = \emptyset$, but by the naturality it also restricts to the relative case. +---PAGE_BREAK--- + +$w_0, \dots, w_{n-1}$. Both $v_i$ and $w_i$ are projected to the same vertex of $\Delta^{n-1}$ which we denote as $u_i$. The prism is given by + +$$ (2.48) \qquad P_I \tilde{\sigma} = \sum_{i=0}^{n-1} (-1)^i (\mathrm{id}_I \times \tilde{\sigma}) |[v_0, \dots, v_i, w_i, \dots, w_{n-1}]. $$ + +Smooth singular version is defined by the same formula. It has the property that + +$$ (2.49) \qquad \partial \circ P_I + P_I \circ \partial = (i_b)_* - (i_a)_*, $$ + +where $i_a: X \simeq \{a\} \times X \subset I \times X$ and $i_b: X \simeq \{b\} \times X \subset I \times X$ are inclusions. + +We also have the property that + +$$ (2.50) \qquad (\mathrm{pr}_X)_* \circ P_I = \partial \circ Q - Q \circ \partial, $$ + +for the map $Q: C_{n-1}(X; E) \to C_{n+1}(X; E)$ defined by + +$$ (2.51) \qquad Q\tilde{\sigma} = \sum_{i=0}^{n-1} \tilde{\sigma}|[u_0, \dots, u_{i-1}, u_i, u_i, u_{i+1}, \dots, u_{n-1}]. $$ + +It particular, for a cycle $t \in Z_{n-1}(X; E)$, $(\mathrm{pr}_X)_* P_I t = \partial Qt$ is a boundary. + +Let $P_I^*: C^n(I \times X; \mathrm{pr}_X^*E) \to C^{n-1}(X; E)$ and $Q^*: C^{n+1}(X; E) \to C^{n-1}(X; E)$ be the duals of $P_I$ and $Q$, respectively, defined as $\langle P_I^*\mu, \sigma \rangle := \langle \mu, P_I\sigma \rangle$ and $\langle Q^*\mu, \sigma \rangle := \langle \mu, Q\sigma \rangle$. They satisfy + +$$ (2.52) \qquad P_I^* \circ \delta + \delta \circ P_I^* = i_b^* - i_a^*, \quad P_I^* \circ \mathrm{pr}_X^* = Q^* \circ \delta - \delta \circ Q^*. $$ + +Moreover, by combining $P_{D^1}$ (where $D^1 = [-1, 1]$) with the map $i: (D^1, S^0) \to (S^1, \{\text{pt}\})$, we define $P_{S^1}: C_{n-1}(X; E) \to C_n(S^1 \times X; \mathrm{pr}_X^*E)$ and $\int: C^n(S^1 \times X; \mathrm{pr}_X^*E) \to C^n(X; E)$ as + +$$ (2.53) \qquad P_{S^1} := i_* \circ P_{D^1}, \qquad \int := P_{S^1}^* = P_{D^1}^* \circ i^*. $$ + +$\int$ satisfies + +$$ (2.54) \qquad \int \circ \delta + \delta \circ \int = 0, \qquad \int \circ \mathrm{pr}_X^* = Q^* \circ \delta - \delta \circ Q^*. $$ + +One can also check that for $\phi_1 \in C^p(S^1 \times X; \mathrm{pr}_X^*E)$ and $\phi_2 \in C^q \in (S^1 \times X; \mathrm{pr}_X^*F)$, + +$$ (2.55) \qquad \int (\phi_1 \cup \phi_2) = (\int \phi_1) \cup (k^*\phi_2) + (-1)^p(k^*\phi_1) \cup (\int \phi_2) $$ + +where $k: \{\text{pt}\} \to S^1$. + +Notice that, if $X$ is a compact $(n-1)$-dimensional manifold and $t \in C_{n-1}(X; \mathrm{Ori}(X))$ is a representative of the fundamental class, then its prism $P_I t \in C_n(I \times X; \mathrm{Ori}(X))$ represents the fundamental chain for $I \times X$ if we use the identification + +$$ \mathrm{pr}_X^* (\mathrm{Ori}(X)) \simeq (\mathrm{Ori}(I \times X)) $$ + +induced by $T(I \times X) = \mathbb{R} \oplus \mathrm{pr}_X^* TX$ and the sign convention (2.35) by this order. +---PAGE_BREAK--- + +**2.5. Generalized differential cohomology theories.** In this subsection we give a brief review of generalized differential cohomology theories, based on the axiomatic framework given in [BS10]. A *differential extension* (also called a *smooth extension*) of a generalized cohomology theory $\{E^*, \delta^*\}$ is a refinement $\hat{E}^*$ of the restriction of $E^*$ to the category of smooth manifolds, which contains differential-geometric data. + +Let $\{E^*, \delta^*\}$ be a generalized cohomology theory. We set + +$$ (2.56) \qquad V_E^\bullet := E^\bullet(\text{pt}) \otimes \mathbb{R}. $$ + +For a manifold $X$, set $\Omega^*(X; V_E^\bullet) := C^\infty(X; \wedge T^*M \otimes_R V_E^\bullet)$ with the $\mathbb{Z}$-grading by the total degree. Let $d: \Omega^*(X; V_E^\bullet) \to \Omega^{*+1}(X; V_E^\bullet)$ be the de Rham differential. We have the natural transformation + +$$ \text{Rham}: \Omega_{\text{clo}}^*(X; V_E^\bullet) \to H^*(X; V_E^\bullet). $$ + +Furthermore, there is a canonical natural transformation of cohomology theories called the *Chern-Dold homomorphism* [Rud98, Chapter II, 3.17] + +$$ \text{ch}: E^*(-) \to H^*(-; V_E^\bullet), $$ + +characterized by the property that, on $\{\text{pt}\}$, + +$$ \text{ch}^k: E^k(\text{pt}) \to H^k(\text{pt}; V_E^\bullet) = E^k(\text{pt}) \otimes \mathbb{R} $$ + +is given by the map $e \mapsto e \otimes 1$. + +**Definition 2.57** (Differential extensions of a cohomology theory, [BS10, Definition 1.1]). A *differential extension* of a generalized cohomology theory $\{E^*, \delta^*\}$ is a quadruple $(\hat{E}, R, I, a)$, where + +• $\hat{E}$ is a contravariant functor $\hat{E}: \text{Mfd}^{\text{OP}} \to \text{Ab}^\mathbb{Z}$. + +• $R, I$ and $a$ are natural transformations + +$$ +\begin{align*} +R: \hat{E}^* &\to \Omega_{\text{clo}}^*(-; V_E^\bullet) \\ +I: \hat{E}^* &\to E^* \\ +a: \Omega^{*-1}(-; V_E^\bullet)/\text{im}(d) &\to \hat{E}^*. +\end{align*} + $$ + +We require the following axioms. + +• $R \circ a = d$. + +• $\text{ch} \circ I = \text{Rham} \circ R$. + +• For all manifolds $X$, the sequence + +$$ E^{*-1}(X) \xrightarrow{\text{ch}} \Omega^{*-1}(M)/\text{im}(d) \xrightarrow{a} \hat{E}(X) \xrightarrow{I} E^*(X) \to 0 $$ + +is exact. + +Such a quadruple $(\hat{E}, R, I, a)$ itself is also called a *generalized differential cohomology theory*. We usually abbreviate the notation and just write a generalized cohomology theory as $\hat{E}^*$. + +*Example 2.58 (Differential characters)*. Here we explain a model for a differential extension of $\mathcal{H}\mathbb{Z}$ given by Cheeger and Simons, in terms of differential characters [CS85]. Actually, our definition of the group $(I\Omega_{\text{DR}}^G)^*$ (Definition 4.11) is analogous to it. We relate them in Subsection 4.4. For later use, we explain the relative version. For another formulation see [BT06]. +---PAGE_BREAK--- + +For a pair of manifolds $(X, A)$ and a nonnegative integer $n$, the group of differential characters $\hat{H}_{\text{CS}}^n(X, A; \mathbb{Z})$ is the abelian group consisting of pairs $(\omega, k)$, where + +• A closed differential form $\omega \in \Omega_{\text{clo}}^n(X, A)$, + +• A group homomorphism$^8$ $k: Z_{\infty,n-1}(X, A; \mathbb{Z}) \to \mathbb{R}/\mathbb{Z}$, + +• $\omega$ and $k$ satisfy the following compatibility condition. For any $c \in C_{\infty,n}(X;\mathbb{Z})$ we have + +$$ (2.59) \qquad k(\partial c) \equiv \langle \omega, c \rangle_X \pmod{\mathbb{Z}}. $$ + +We have homomorphisms + +$$ R_{\text{CS}}: \hat{H}_{\text{CS}}^{n}(X, A; \mathbb{Z}) \to \Omega_{\text{clo}}^{n}(X, A), \quad (\omega, k) \mapsto \omega $$ + +$$ a_{\text{CS}} : \Omega^{n-1}(X, A)/\operatorname{Im}(d) \to \hat{H}_{\text{CS}}^{n}(X, A; \mathbb{Z}), \quad \alpha \mapsto (d\alpha, \alpha). $$ + +and the quotient map gives + +$$ I_{\text{CS}}: \hat{H}_{\text{CS}}^{n}(X, A; \mathbb{Z}) \to \hat{H}_{\text{CS}}^{n}(X, A; \mathbb{Z}) / \operatorname{Im}(a_{\text{CS}}) \simeq H^{n}(X, A; \mathbb{Z}). $$ + +The quadruple $(\hat{H}_{\text{CS}}^*, R_{\text{CS}}, I_{\text{CS}}, a_{\text{CS}})$ is a differential extension of $HZ$. + +We can also consider the differential refinement of $S^1$-integrations maps in (2.9). We have the $S^1$-integration map for differential forms, + +$$ (2.60) \qquad \int : \Omega^{n+1}(S^1 \times X; V_E^\bullet) \to \Omega^n(X; V_E^\bullet) $$ + +for any manifold $X$, which realizes the $S^1$-integration map in de Rham cohomology. The sign is defined so that + +$$ (2.61) \qquad \int \mathrm{pr}_{S^1}^* \tau_{S^1} \wedge \mathrm{pr}_X^* \omega_X = \omega_X $$ + +for $\omega_X \in \Omega^*(X; V)$ and $\tau_{S^1} \in \Omega_{\text{clo}}^1(S^1)$ which represents the fundamental class of $S^1$ for the standard orientation. + +**Definition 2.62** (Differential extensions with $S^1$-integrations, [BS10, Definition 1.3]). + +A differential extension with integration of $E^*$ is a quintuple $(\hat{E}, R, I, a, \int)$, where $(\hat{E}, R, I, a)$ is a differential extension of $E^*$ and $\int$ is a natural transformation + +$$ \int : \hat{E}^{*+1}(S^1 \times -) \to \hat{E}^* $$ + +such that + +• $\int \circ (t \times \text{id})^* = -\int$, where $t: S^1 \to S^1$ is given by $t(x) = -x$ for $x \in [-1, 1]/\{-1, 1\} = S^1$. + +• $\int \circ (p_{S^1} \times \text{id})^* = 0$. + +$^8$Precisely speaking in [CS85] they use normalized smooth singular cubic chains, but we can also work in terms of the usual smooth singular chains as in [BT06]. +---PAGE_BREAK--- + +• The diagram + +(2.63) + +commutes for all manifolds $X$. + +**2.6.** $\langle k \rangle$-manifolds. In this paper, we need to deal with manifolds with corners. There are some variants in the definition among the literatures. In this paper the appropriate notion is $\langle k \rangle$-manifolds defined in [J68], which we recall here. + +A manifold with corners of dimension $n$ is a paracompact topological space $M$ with a local coordinate system $\{V_i, \varphi_i\}_{i \in I}$, where $V_i$ is an open subset of $M$ for each $i$ with $M = \cup_i V_i$, + +$$ \varphi_i: V_i \to \mathbb{R}_{\le 0}^n := (-\infty, 0]^n $$ + +is a homeomorphism onto an open subset of $\mathbb{R}_{\le 0}^n$ for each $i$, and + +$$ \varphi_i \varphi_j^{-1}: \varphi_j(V_i \cap V_j) \to \varphi_i(V_i \cap V_j) $$ + +is a diffeomorphism for all pairs $(i, j)$. For a point $x \in M$, let $c(x)$ be the number of zeros in a local chart, which is well-defined. A manifold with corners is called a *manifold with faces* if each $x \in M$ belongs to the closure of $c(x)$ different components of $\{p \in M \mid c(p) = 1\}^9$. For a manifold with faces $M$ of dimension $n$, the closure of a connected component of $\{p \in M \mid c(p) = 1\}$ is called a *connected face* of $M$, which has the induced structure of an $(n-1)$-dimensional manifold with faces. Any union of pairwise disjoint connected faces is called a *face* of $M$. + +**Definition 2.64** ($(\langle k \rangle$-manifolds, [J68, Definition 1]). Let $k$ be a nonnegative integer. A $\langle k \rangle$-manifold is a manifold with faces $M$ together with an $k$-tuple $(\partial_0 M, \partial_1 M, \dots, \partial_{k-1} M)$ of faces of $M$, satisfying + +$$ (1) \quad \partial_0 M \cup \dots \cup \partial_{k-1} M = \partial M. $$ + +$$ (2) \quad \partial_i M \cap \partial_j M \text{ is a face of } \partial_i M \text{ and of } \partial_j M \text{ for } i \neq j. $$ + +In particular, a $\langle 0 \rangle$-manifold is equivalent to a manifold without boundary, and a $\langle 1 \rangle$-manifold is equivalent to a manifold with boundary. See Figure 1 for $k=2$. Note that each $\partial_i M$ can be empty. For $I = \{0 \le i_1 < \dots < i_m \le k-1\}$, the intersection $\partial_I M := \partial_{i_1} M \cap \dots \cap \partial_{i_m} M$ is called an $\langle k-m \rangle$-face of $M$. It has the induced structure of an $\langle k-m \rangle$-manifold so that the order of the labels of the faces are preserved. + +In this paper we are particularly interested in $\langle 0 \rangle$, $\langle 1 \rangle$ and $\langle 2 \rangle$-manifolds. + +**Definition 2.65** (Collar structures, Figure 2). For a $\langle k \rangle$-manifold $M$, a collar structure on $M$ is defined inductively in $k$, as follows. + +⁹For example, this excludes the case of the “teardrop”, the 2-dimensional disk with a corner. +---PAGE_BREAK--- + +FIGURE 1. A ⟨2⟩-manifold + +* For $k=0$, no other data is required. + +* For $k=1$, it is a triple $(U, \phi, \epsilon)$, where $U$ is an open neighborhood of $\partial M$, $\epsilon > 0$ is a positive real number, and $\phi$ is a diffeomorphism + +$$ \phi: (-\epsilon, 0] \times \partial M \simeq U. $$ + +* For general $k$, it consists of + + - For each nonempty subset $I$ of $\{0, 1, \dots, k-1\}$, a collar structure on $\partial_I M$ (which is a $\langle k-m \rangle$-manifold). + + - A triple $(U_i, \phi_i, \epsilon_i)$ for each $0 \le i \le k-1$, where $U_i$ is an open neighborhood of $\partial_i M$, $\epsilon_i > 0$ is a positive real number, and $\phi_i$ is a diffeomorphism + +$$ \phi_i: (-\epsilon_i, 0] \times \partial_i M \simeq U_i. $$ + + - The above data should satisfy the following compatibility. For each pair $(i,j)$ with $0 \le i < j \le k-1$, the collar structure on $\partial_i M$ and $\partial_j M$ has the data of diffeomorphism + +$$ \phi_{ij}: (-\epsilon_{ij}, 0] \times \partial_{\{i,j\}} M \simeq U_{ij}, \\ +\phi_{ji}: (-\epsilon_{ji}, 0] \times \partial_{\{i,j\}} M \simeq U_{ji}, $$ + +where $U_{ij}$ is an open neighborhood of $\partial_{\{i,j\}} M$ in $\partial_i M$ and $U_{ji}$ is that of $\partial_{\{i,j\}} M$ in $\partial_j M$. Let $\epsilon := \min\{\epsilon_{ij}, \epsilon_{ji}, \epsilon_i, \epsilon_j\}$. We have two maps which are diffeomorphisms onto their image, + +$$ (2.66) \quad \begin{aligned} & (-\epsilon, 0] \times (-\epsilon, 0] \times \partial_{\{i,j\}} M \to U_i \cap U_j, \\ & (s, t, x) \mapsto \phi_i(s, \phi_{ij}(t, x)) \\ & (s, t, x) \mapsto \phi_j(t, \phi_{ji}(s, x)). \end{aligned} $$ + +We require that those maps agree. + +In the above, the explicit values of $\epsilon$, $\epsilon_i$ and $\epsilon_{ij}$'s are irrelevant. If a collar structure is obtained by replacing some of the constants in another collar structure smaller, we regard them as the same collar structure. + +For a $\langle k \rangle$-manifold with a collar structure, its $\langle k-m \rangle$-faces become $\langle m \rangle$-manifold with collar structures. We use the obvious isomorphism relations between $\langle k \rangle$-manifolds with collar structures. + +**Definition 2.67** (Bordism relations of $\langle k \rangle$-manifolds with collar structures). Let $M$ be an $(n-1)$-dimensional $\langle k-1 \rangle$-manifold with a collar structure and $W$ be an $n$-dimensional $\langle k \rangle$-manifold with a collar structure. We say that $M$ +---PAGE_BREAK--- + +FIGURE 2. A collar structure + +bounds $W$ if we are given an isomorphism $\varphi: \partial_0 W \simeq M$ of $\langle k-1 \rangle$-manifold +with a collar structure. + +For maps from a $\langle k \rangle$-manifold with a collar structure to another space, +we want to consider those which are constant in the collar coordinate, as +follows. + +**Definition 2.68** (Maps which are compatible with collar structures). Let $M$ be a $\langle k \rangle$-manifold with a collar structure and $X$ be a topological space. We say that a continuous map $f: M \to X$ is *compatible with the collar structure* if, for each $I \subset \{0, 1, \dots, k-1\}$ and $j \in \{0, 1, \dots, k-1\} \setminus I$ the following condition is satisfied. Let $\phi_{Ij}: (-\epsilon_{Ij}, 0] \times \partial_{I \cup \{j\}} M \to U_{Ij}$ be the diffeomorphism in the data of collar structure, where $U_{Ij}$ is an open neighborhood of $\partial_{Ij}M$ in $\partial IM$. We require that + +$$f|_{\partial I \cup \{j\} M} \circ \text{pr}_{\partial I \cup \{j\} M} = f|_{U_{Ij}} \circ \phi_{Ij} : (-\epsilon_{Ij}, 0] \times \partial I \cup \{j\} M \to X.$$ + +**Definition 2.69.** Let $M$ be an $(n-1)$-dimensional $\langle k-1 \rangle$-manifold with a collar structure and $W$ be an $n$-dimensional $\langle k \rangle$-manifold with a collar structure. Assume $M$ bounds $W$ in the sense of Definition 2.67, with the isomorphism $\varphi: \partial_0 W \simeq M$. A representative $t_W \in C_{\infty,n}(W; \text{Ori}(W))$ of the fundamental class of $W$ extends a representative $t_M \in C_{\infty,n-1}(M; \text{Ori}(M))$ of the fundamental class of $M$ if we have (recall our convention on the orientation bundles in (2.39)) + +$$\partial t_W - \varphi^* t_M \in C_{\infty,n-1}(\partial W \setminus \text{int}(\partial_0 W); \text{Ori}(\partial W)).$$ + +Here $\text{int}(\partial_0 W)$ is the interior of $\partial_0 W$. + +The following lemma will be used in the proof of Proposition 4.67. + +**Lemma 2.70.** Let $M$ be an $(n-1)$-dimensional $\langle k-1 \rangle$-manifold with a collar structure and $W$ be an $n$-dimensional $\langle k \rangle$-manifold with a collar structure. Assume $M$ bounds $W$ in the sense of Definition 2.67. Given any representative $t_M \in C_{\infty,n-1}(M; \text{Ori}(M))$ of the fundamental class of $M$, there exists a representative $t_W \in C_{\infty,n}(W; \text{Ori}(W))$ of the fundamental class of $W$ which extends $t_M$ in the sense of Definition 2.69. +---PAGE_BREAK--- + +*Proof.* We have the excision isomorphism, + +$$ +(2.71) \qquad H_{n-1}^{\text{sing},\infty}(M, \partial M; \text{Ori}(M)) \simeq H_{n-1}^{\text{sing},\infty}(\partial W, \partial W \setminus \text{int}(\partial_0 W); \text{Ori}(\partial W)). +$$ + +Recall that this is induced by the map at the chain level, + +$$ +\begin{align*} +& C_{\infty,n-1}(M; \mathrm{Ori}(M))/C_{\infty,n-1}(\partial M; \mathrm{Ori}(\partial M)) \\ +& \to C_{\infty,n-1}(\partial W; \mathrm{Ori}(\partial W))/C_{\infty,n-1}(\partial W \setminus \mathrm{int}(\partial_0 W); \mathrm{Ori}(\partial W)), +\end{align*} +$$ + +induced by the inclusion $C_{\infty,n-1}(M; \mathrm{Ori}(M)) \hookrightarrow C_{\infty,n-1}(\partial W; \mathrm{Ori}(\partial W))$. + +On the other hand we have the map $H_{n-1}^{\text{sing},\infty}(\partial W; \text{Ori}(\partial W)) \to H_{n-1}^{\text{sing},\infty}(\partial W, \partial W \setminus \text{int}(\partial_0 W); \text{Ori}(\partial W))$, and the fundamental class $[\partial W]$ for $\partial W$ maps to the fundamental class $[M]$ for $M$ under this map composed with the isomorphism (2.71). + +Take any representative $t'_W \in C_{\infty,n}(W; \mathrm{Ori}(W))$ of the fundamental class of $W$. Then $\partial t'_W \in Z_{\infty,n-1}(\partial W; \mathrm{Ori}(\partial W))$ represents $[\partial W]$. By the above consideration, this means that there exists a chain $s \in C_{\infty,n}(\partial W; \mathrm{Ori}(\partial W))$ with + +$$ +\partial t'_{W} - \varphi^{*} t_{M} + \partial s \in C_{\infty, n-1}(\partial W \setminus \text{int}(\partial_{0} W); \text{Ori}(\partial W)). +$$ + +Then, the element $t_W := t'_W + s$ satisfies the desired property. $\square$ + +For a ⟨1⟩-manifold *M* with a collar structure, its tangent bundle *TM* is identified with ℝ⊕T∂*M*, via *φ*, on the collar *U*. For a ⟨2⟩-manifold *M* with a collar structure, *TM* is identified with ℝ⊕T∂ᵢ*M*, via *φ*ᵢ, on the collar *U*ᵢ for *i* = 0, 1, and is identified with ℝ⊕ Real T(∂₀*M* ∩ ∂₁*M*) on the image of the map (2.66). + +3. GEOMETRIC STABLE TANGENTIAL G-STRUCTURES + +In this section, we set up the notion of geometric stable tangential G- +structures on manifolds. + +**3.1. Geometric stable tangential G-structures.** Let $G = \{G_d, s_d, \rho_d\}_{d \in \mathbb{N}}$ +be as in Subsection 2.3. We work with $\langle k \rangle$-manifolds with a collar structure +for $k = 0, 1, 2$ explained in Subsection 2.6. We are going to require that all +the additional structure on such manifolds are compatible with the collar +structures. + +**Definition 3.1** (Geometric stable G-structures on vector bundles). Let $V$ be a real vector bundle of rank $n$ over a manifold $M$. A geometric stable G-structure on $V$ is a quadruple $(d, P, \nabla, \psi)$, where $d \ge n$ is an integer, $(P, \nabla)$ is a principal $G_d$-bundle with connection over $M$ and $\psi: P \times_{\rho_d} \mathbb{R}^d \simeq \mathbb{R}^{d-n} \oplus V$ is an isomorphism of vector bundles over $M$. + +We use the obvious isomorphism relation between geometric stable G- +structures on vector bundles. + +Now we define geometric stable tangential *G*-structures on manifolds. +They are essentially geometric stable *G*-structures on tangent bundles, but +we need to require compatibility with collar structures. The precise defini- +tion is the following. +---PAGE_BREAK--- + +**Definition 3.2 (Geometric stable tangential G-structures).** Let $M$ be a $\langle k \rangle$-manifold with a collar structure. A geometric stable tangential G-structure on $M$ consists of the following data. + +• An integer $d \ge \dim M$. + +• A geometric stable G-structure $g_I = (d, P_I, \nabla_I, \psi_I)$ on the tangent bundles $T(\partial_I M)$ over $\partial_I M$ for each $I \subset \{0, 1, \dots, k-1\}$. + +• For each $I \subset \{0, 1, \dots, k-1\}$ and $j \in \{0, 1, \dots, k-1\} \setminus I$, an isomorphism, setting $I_j := I \cup \{j\}$, + +$$ \varphi_{Ij}: (g_{Ij})_{(-\epsilon_{Ij}, 0]} \simeq \phi_{Ij}^* g|_{U_{Ij}} $$ + +of geometric stable G-structures on $T((-\epsilon_{Ij}, 0] \times \partial_{Ij}M)$. Here $\phi_{Ij}: (-\epsilon_{Ij}, 0] \times \partial_{Ij}M \to U_{Ij}$ is the diffeomorphism in the data of collar structure, where $U_{Ij}$ is an open neighborhood of $\partial_{Ij}M$ in $\partial_I M$. We denote by $(g_{Ij})_{(-\epsilon_{Ij}, 0]}$ the geometric stable G-structure on $T((-\epsilon_{Ij}, 0] \times \partial_{Ij}M)$ defined as $(g_{Ij})_{(-\epsilon_{Ij}, 0]} := (d, \text{pr}_{\partial_{Ij}M}^* P_{Ij}, \text{pr}_{\partial_{Ij}M}^*\nabla_{Ij}, \text{pr}_{\partial_{Ij}M}^*\psi_{Ij})$, using the identification $\mathbb{R}^{d-n-1} \oplus T((-\epsilon_{Ij}, 0] \times \partial_{Ij}M) \simeq \text{pr}_{\partial_{Ij}M}^*(\mathbb{R}^{d-n} \oplus T\partial_{Ij}M)$. + +In the following, we just denote a geometric stable tangential *G*-structure by *g* or *g* = (d, *P*, ∇, ψ), with the understanding that all the other data are included. We use the obvious isomorphism relations between geometric stable tangential *G*-structures. By definition, a geometric stable tangential *G*-structure *g* on *M* induces geometric stable tangential *G*-structures, denoted by ∂_I*g*, on its ⟨*k* − *m*⟩-faces ∂_I*M*. + +The following notation, which was also used in Definition 3.2, is useful. Let $M$ be an $n$-dimensional $\langle k \rangle$-manifold and $J \subset \mathbb{R}$ be an interval. Given a geometric stable tangential $G$-structure $g = (d, P, \nabla, \psi)$ with $d \ge n+1$ on $M$, we define the geometric stable tangential $G$-structure $g_J$ on $J \times M$ by + +$$ (3.3) \qquad g_J := (d, \mathrm{pr}_M^* P, \mathrm{pr}_M^* \nabla, \mathrm{pr}_M^* \psi), $$ + +using the identification $\mathbb{R}^{d-n-1} \oplus T(J \times M) \simeq \mathrm{pr}_M^*(\mathbb{R}^{d-n-1} \oplus \mathbb{R} \oplus TM) = \mathrm{pr}_M^*(\mathbb{R}^{d-n} \oplus TM)$. + +**Definition 3.4 (Opposite geometric stable tangential G-structure).** Let $M$ be an $n$-dimensional $\langle k \rangle$-manifold with a collar structure. Given a geometric stable tangential $G$-structure $g = (d, P, \nabla, \psi)$ on $M$ with $d > n$, we define its opposite geometric stable tangential $G$-structure $g_{op}$ by $g_{op} := (d, P, \nabla, (\mathrm{id}_{\mathbb{R}^{d-n-1}} \oplus -\mathrm{id}_{\mathbb{R}} \oplus \mathrm{id}_{TM}) \circ \psi)^{10}$. + +**Definition 3.5 (Bordism relations of geometric stable tangential G-structures).** Let $M$ be an $(n-1)$-dimensional $\langle k-1 \rangle$-manifold with a collar structure and $W$ be an $n$-dimensional $\langle k \rangle$-manifold with a collar structure, bounded by $M$ in the sense of Definition 2.67 with the isomorphism $\varphi: \partial_0 W \simeq M$. Assume we are given geometric stable tangential $G$-structures $g_W$ and $g_M$ on $W$ and $M$, respectively. We say that $g_M$ bounds $g_W$ if we are given an + +¹⁰Of course, in the case of $k=1,2$, geometric stable *G*-structures on the tangent bundles of $\langle k-m \rangle$-faces are also changed correspondingly. Similar remarks apply to Definition 3.6. +---PAGE_BREAK--- + +isomorphism of geometric stable tangential $G$-structures $\partial_0 g_W \simeq \varphi^* g_M$ on $\partial_0 W$. + +**Definition 3.6** (Stabilizations of geometric stable tangential $G$-structures). For a geometric stable tangential $G$-structure $g = (d, P, \nabla, \psi)$ on a $\langle k \rangle$-manifold $M$ with a collar structure and a positive integer $l$, we define the $l$-fold stabilization $g(l)$ of $g$ to be the geometric stable tangential $G$-structure on $M$ given by $g(l) := (d+l, P(l)) := P \times_{s_{d+l,d}} G_{d+l}, \nabla(l), \psi(l))$, where $s_{d+l,d} := s_{d+l-1} \circ \cdots \circ s_d$, and $\nabla(l)$ and $\psi(l)$ are naturally induced on $P(l)$ from $\nabla$ and $\psi$, respectively. + +**Definition 3.7** (Geometric smooth stable tangential $G$-cycles). Let $(X, A)$ be a pair of manifolds. A geometric smooth stable tangential $G$-cycle of dimension $n$ over $(X, A)$ is a triple $(M, g, f)$, where $M$ is an $n$-dimensional compact $\langle 1 \rangle$-manifold with a collar structure, $g$ is a geometric stable tangential $G$-structure on $M$ and $f$ is a smooth map $f \in C^\infty((M, \partial M), (X, A))$ which is compatible with the collar structure (Definition 2.68). We define isomorphisms between two geometric smooth stable tangential $G$-cycles in an obvious way. + +We introduce the equivalence relation $\sim$ on geometric smooth stable tangential $G$-cycles of dimension $n$ over $(X, A)$, generated by + +• $(M, g, f) \sim (M', g', f')$ if they are isomorphic, + +• $(M, g, f) \sim (M, g(1), f)$, and + +• $(M, g, f) \sqcup (M, g_{\text{op}}, f) \sim \emptyset$. + +The set of equivalence classes under $\sim$ is denoted by $C_{\infty,n}^G(X, A)$. The isomorphism class of $(M, g, f)$ is denoted by $[M, g, f]$. We introduce an abelian group structure on $C_{\infty,n}^G(X, A)$ by disjoint union. + +**Definition 3.8** (Bordism data of geometric smooth stable tangential $G$-cycles, Figure 3). Let $(M, g_M, f_M)$ be a geometric smooth stable tangential $G$-cycle of dimension $(n-1)$ over $(X, A)$. A bordism data of $(M, g_M, f_M)$ is a triple $(W, g_W, f_W)$, where $W$ is an $n$-dimensional $\langle 2 \rangle$-manifold with a collar structure which is bounded by $M$ with the isomorphism $\varphi: \partial_0 W \simeq M$ (Definition 2.67), $g_W$ is a geometric stable tangential $G$-structure on $W$ which is bounded by $g_M$ (Definition 3.5) and $f_W \in C^\infty((W, \partial_1 W), (X, A))$ is a map which is compatible with the collar structure and satisfies $f_W|_{\partial_0 W} = f_M \circ \varphi$. + +*Example 3.9.* For a geometric smooth stable tangential $G$-cycle $(M, g, f)$ over $(X, A)$, we can regard $([0,1] \times M, g|_{[0,1]}, f \circ \text{pr}_M)$ (see (3.3)) as a bordism data of $(M, g, f) \sqcup (M, g_{\text{op}}, f)$ in the obvious way. + +In Subsection 4.1.3, we define singular models for the Anderson duals to $G$-bordism theories. For that purpose, we need to equip manifolds with the data of singular chains. + +**Definition 3.10** (Geometric singular stable tangential $G$-cycles). (1) Let $(X, A)$ be a pair of topological spaces. A geometric singular stable tangential $G$-cycle of dimension $n$ over $(X, A)$ is a quadruple $(M, g, t, f)$, where $M$ is an $n$-dimensional compact $\langle 1 \rangle$-manifold with a collar structure, $g$ is a geometric stable tangential $G$-structure on $M$, $t$ is a smooth singular chain $t \in C_{\infty,n}(M; \text{Ori}(M))$ which +---PAGE_BREAK--- + +FIGURE 3. A bordism data + +represents the fundamental class $[M] \in H_n(M, \partial M; \text{Ori}(M))$, and +$f \in \text{Map}((M, \partial M), (X, A))$ is a map compatible with the collar +structure. + +We introduce the equivalence relation on geometric singular stable +tangential *G*-cycles of dimension *n* over (X, A), generated by + +• $(M, g, t, f) \sim (M', g', t', f')$ if they are isomorphic, + +• $(M, g, t, f) \sim (M, g(1), t, f)$, and + +• $(M, g, t, f) \sqcup (M, g_{\text{op}}, t, f) \sim \emptyset$. + +The set of equivalence classes under $\sim$ is denoted by $C_{\Delta,n}^G(X, A)$. The isomorphism class of $(M, g, t, f)$ is denoted by $[M, g, t, f]$. We introduce an abelian group structure on $C_{\Delta,n}^G(X, A)$ by disjoint union. + +(2) When $(X, A)$ is a pair of manifolds, geometric smooth singular stable tangential $G$-cycle of dimension $n$ over $(X, A)$ is a quadruple $(M, g, t, f)$ as above, and in addition, we require that $f$ is smooth. We define the abelian group $C_{\Delta,\infty,n}^G(X, A)$ similarly. + +We have the boundary map, + +$$ +(3.11) \qquad \partial: C_{\Delta,n}^{G}(X,A) \to C_{\Delta,n-1}^{G}(A), +$$ + +by mapping $(M, g, t, f)$ to $(\partial M, \partial g, \partial t, f|_{\partial M})$. + +For a pair of manifolds $(X, A)$, we have + +$$ +C_{\Delta,\infty,n}^G(X,A) \subset C_{\Delta,n}^G(X,A) +$$ + +and we have the forgetful map, + +$$ +(3.12) \qquad \text{fgt}: C_{\Delta,\infty,n}^{G}(X,A) \to C_{\infty,n}^{G}(X,A), \quad [M,g,t,f] \mapsto [M,g,f]. +$$ + +**Definition 3.13** (Bordism data of geometric singular stable tangential *G*-cycles). Let (*M*, *g**M*, *t**M*, *f**M*) be a geometric singular stable tangential *G*-cycle of dimension (*n*−1) over (*X*, *A*). A *bordism data* of (*M*, *g**M*, *t**M*, *f**M*) is a quadruple (*W*, *g**W*, *t**W*, *f**W*), where *W* is an *n*-dimensional ⟨2⟩-manifold with a collar structure which is bounded by *M* with the isomorphism ϕ: ∂₀*W* ≅ *M* (Definition 2.67), *g**W* is a geometric stable tangential *G*-structure on *W* which is bounded by *g**M* (Definition 3.5), *t**W* extends *t**M* in the sense of Definition 2.69, and *f**W* ∈ Map((*W*, ∂₁*W*), (*X*, *A*)) is a map which is compatible with the collar structure and satisfies *f**W*|∂₀*W* = *f**M* ◦ ϕ. +---PAGE_BREAK--- + +We define the bordism data for geometric smooth singular stable tangential $G$-cycles similarly. + +**3.2. Geometric Pontryagin-Thom construction.** In this subsection, we introduce a geometric refinement of the Pontryagin-Thom construction. We only use it in the proof of the main theorem in Subsection 5.2, so the reader who is only interested in the results may skip this subsection and go directly to the next section. + +The setting is the following. Let $d \in \mathbb{Z}_{\ge 0}$, and $G_d$ be a Lie group with a homomorphism $\rho_d: G_d \to O(d, \mathbb{R})$. Assume we are given a smooth principal $G_d$-bundle with a $G_d$-connection $(P, \nabla)$ on a (possibly noncompact) smooth manifold $B$ without boundary. We set $\mathcal{V}_d := P \times_{\rho_d} \mathbb{R}^d$, and assume we are given a rank-$q$ real vector bundle $\pi: W_q \to B$ with inner product, equipped with a fixed isomorphism + +$$ (3.14) \qquad \mathcal{V}_d \oplus W_q \simeq \mathbb{R}^{d+q} $$ + +of vector bundles. We also fix a smooth cocycle representative of the Thom class $\tau \in Z^\mu_\infty(\text{Thom}_\infty(W_q); \text{Ori}(\pi))$ (recall our notation (1.17)). + +**Definition 3.15.** Let $n$ be a nonnegative integer and $(X, A)$ be a pair of topological spaces. We say that a geometric singular stable tangential fr-cycle $(M, g, t, f)$ of dimension $(n+q)$ over $(X, A) \times \text{Thom}_\infty(W_q)$ is nice with a tubular neighborhood structure $\pi_f$ if the following three conditions are satisfied. + +(A) The map $\text{pr}_{D(W_q)} \circ f: M \to D(W_q)$ is smooth and transverse to the zero section of $W_q$. We set $M_f := f^{-1}(X \times B)$, which is a compact $n$-dimensional $\langle 1 \rangle$-manifold with a collar structure. + +(B) $\nu := f^{-1}(X \times \check{D}(W_q))$ is a tubular neighborhood of $M_f$ in $M$ with a smooth disk bundle structure $\pi_f: \bar{\nu} \to M_f$ such that the diagram commutes, + +and the induced map $\bar{\nu} \to (\text{pr}_B \circ f|_{M_f})^*D(W_q)$ is an isomorphism of disk bundles. We require that the disk bundle structure is compatible with the collar structure at $\partial M$ in the obvious way. + +(C) $t \in C_{\infty,n+q}(M; \text{Ori}(M))$ restricts to a representative of a fundamental class $t_{\bar{\nu}} \in C_{\infty,n+q}(\bar{\nu}; \text{Ori}(\bar{\nu}))$, i.e., we have $t - t_{\bar{\nu}} \in C_{\infty,n+q}(M \setminus \nu; \text{Ori}(M))$. + +If $(X, A)$ is a pair of manifolds, we define the niceness with a tubular neighborhood structure for a geometric stable tangential fr-cycle $(M, g, f)$ of dimension $(n+q)$ over $(X, A) \times \text{Thom}_\infty(W_q)$ in the same way, just by forgetting the Condition (C) on the fundamental cycle. + +See Figure 4 for the situation. Remark that any element in $\mathcal{C}_{\Delta,n}^G((X, A) \times \text{Thom}_\infty(W_q))$ is bordant to a nice one, i.e., we can take a bordism data (Definition 3.13) of the opposite and a nice one (use Lemma 2.70). +---PAGE_BREAK--- + +FIGURE 4. Before the geo- +metric Pontryagin-Thom con- +struction (A = ∅) + +FIGURE 5. After the geo- +metric Pontryagin-Thom con- +struction (A = ∅) + +For a nice $(M, g, t, f)$ with a tubular neighborhood structure $\pi_f$ as above, +we equip the submanifold $M_f = f^{-1}(X \times B)$ with a geometric stable +tangential $G_d$-structure$^{11}$, denoted by $g_{f^*\nabla}$, as follows. The transversal- +ity condition (A) implies that, using $\pi_f$, we have a canonical identification +$TM_f \oplus f^*W_q \simeq TM|_{M_f}$ over $M_f$. Using the stable framing $g$ on $TM$, for +sufficiently large $k \in \mathbb{Z}_{\ge 0}$ we get an isomorphism of vector bundles over $M_f$, + +$$ +(3.17) \qquad \mathcal{A}: \mathbb{R}^k \oplus TM_f \oplus (f|_{M_f})^* W_q \simeq \mathbb{R}^{n+q+k}. +$$ + +On the other hand, by (3.14) we have an isomorphism of vector bundles over $M_f$, + +$$ +(3.18) \qquad \mathcal{B} : (f|_{M_f})^* \mathcal{V}_d \oplus (f|_{M_f})^* \mathcal{W}_q \simeq \underline{\mathbb{R}}^{d+q}. +$$ + +We also define $r : \underline{\mathbb{R}}^{d+q} \to \underline{\mathbb{R}}^{d+q}$ as + +$$ +(3.19) \qquad r = \left[ \frac{(-1)^{q+k+n} \mathrm{id}_{\underline{\mathbb{R}}^d}}{0} \mid \frac{0}{(-1)^{k+n} \mathrm{id}_{\underline{\mathbb{R}}^q}} \right]. +$$ + +Combining (3.17), (3.18) and (3.19), we define the isomorphism $\psi(M, g, f, d)$ +by the following map, + +$$ +(3.20) \qquad \begin{aligned}[t] +\psi(M,g,f,d) : \mathbb{R}^{n+q+k} &\oplus (f|_{M_f})^* \mathcal{V}_d \simeq \mathbb{R}^{d+q+k} \oplus TM_f \\ +(\mathcal{A}(x,\xi,w),v) &\mapsto (r\mathcal{B}(v,w),x,\xi) +\end{aligned} +$$ + +for $v \in (f|_{M_f})^*V_d$, $w \in (f|_{M_f})^*W_q$, $x \in \mathbb{R}^k$ and $\xi \in TM_f$. The role +of the sign factors $(-1)^{k+n}$ and $(-1)^{q+k+n}$ in $r$ will be explained below. +The principal $G_d$ bundle $(f|_{M_f})^*P$ over $M_f$ associates the vector bundle +$\mathbb{R}^{n+q+k} \oplus (f|_{M_f})^*V_d$ (see footnote 11), so the quadruple + +$$ +g_{f^*\nabla} := (n+d+q+k, (f|_{M_f})^*P, (f|_{M_f})^*\nabla, \psi(M, g, f, d)) +$$ + +11Here we regard $G_d$ as a sequence $G_d := \{G_{d'}, s_{d'}, \rho_{d'}\}_{d'}$ with $G_{d'} = G_d$, $\rho_{d'} = \text{incl} \circ \rho_d$ +and $s_{d'} = \text{id}$ for $d' \ge d$. +---PAGE_BREAK--- + +is a geometric stable tangential $G_d$-structure (Definition 3.2) on $M_f$ (Figure 5). + +Now we construct a representative of the fundamental class for $M_f$ as +follows. Notice that we have the following canonical isomorphism of vector +bundles over $\bar{\nu}$. + +$$T\bar{\nu} \simeq \pi_f^*(TM_f \oplus \pi_f).$$ + +Using this isomorphism and our convention (2.35) (2.36), we identify the +orientation bundles as + +$$\mathrm{Ori}(\bar{\nu}) \simeq \pi_f^*(\mathrm{Ori}(M_f) \otimes \mathrm{Ori}(\pi_f)).$$ + +Since $(\mathrm{pr}_{D(W_q)} \circ f)^*\tau \in Z^\infty_\infty(\bar{\nu}, \partial\bar{\nu}; \pi_f^*\mathrm{Ori}(\pi_f))$ is a Thom cocycle for the disk bundle $\pi_f: \nu \to M_f$, the smooth singular chain + +$$ (3.21) \qquad t_{M_f, \tau} := (\pi_f)_*((\mathrm{pr}_{D(W_q)} \circ f)^*\tau \cap t_{\bar{\nu}}) \in C_{\infty,n}(M_f; \mathrm{Ori}(M_f)) $$ + +is a representative of the fundamental class of $M_f$. Thus we get a geomet- +ric singular stable tangential $G_d$-cycle ($M_f, g_{f^*\nabla}, t_{M_f, \tau}, \mathrm{pr}_{(X,A)} \circ f|_{M_f}$) of +dimension $n$ over $(X, A)$. + +**Definition 3.22** (Geometric Pontryagin-Thom construction). In the above settings, for a nice geometric singular stable tangential fr-cycle $(M, g, t, f)$ with a tubular neighborhood structure $\pi_f$ of dimension $(n+q)$ over $(X, A) \times \text{Thom}_\infty(W_q)$, we define a geometric singular stable tangential $G_d$-cycle $\text{PT}_{\nabla,\tau}(M, g, t, f)$ of dimension $n$ over $(X, A)$ by + +$$ (3.23) \qquad \text{PT}_{\nabla,\tau}(M,g,t,f) := (M_f, g_{f^*\nabla}, t_{M_f,\tau}, \text{pr}_{(X,A)} \circ f|_{M_f}). $$ + +When $G_d$ is a part of a sequence $G = \{G_d', \rho_d', s_d'\}_{d' \in \mathbb{Z}_{\ge 0}}$ as in Subsection 2.3, we also regard it as a geometric singular stable tangential $G$-cycle of dimension $n$ over $(X, A)$. + +If $(X, A)$ is a pair of manifolds, for a nice geometric stable tangential +fr-cycle $(M, g, f)$ with a tubular neighborhood structure $\pi_f$ of dimension +$(n + q)$ over $(X, A) \times \text{Thom}_\infty(W_q)$, we define a geometric stable tangential +$G_d$-cycle $\text{PT}_\nabla(M, g, f)$ of dimension $n$ over $(X, A)$ in the same way, just by +forgetting the fundamental cycles. + +Now let us explain the reason for introducing the factor *r* in (3.20). Let $\psi(M, g, f, d)(1)$ be the 1-fold stabilization of $\psi(M, g, f, d)$ (Definition 3.6). The next lemma shows that, under our sign convention, the geometric Pontryagin-Thom construction is consistent, up to homotopy of $\psi(M, g, f, d)$, with increasing *k*, *q* and *d*. Moreover it preserves bordism relations. + +**Lemma 3.24.** (1) If we change the stable framing $\psi^{\mathrm{fr}} : \mathbb{R}^k \oplus TM \rightarrow \mathbb{R}^{k+n+q}$ to its 1-fold stabilization $\psi^{\mathrm{fr}}(1) : \mathbb{R}^{k+1} \oplus TM \rightarrow \mathbb{R}^{k+1+n+q}$, +the isomorphisms $\psi(M,g(1),f,d)$ and $\psi(M,g,f,d)(1)$ are homotopic. + +(2) Suppose we change $W_q$ to $W_{q+1} = W_q \oplus \mathbb{R}$ and take $(M', g', f')$ with $\dim M' = \dim M + 1$ in such a way that there is a diffeomorphism $M' \to M$, and near $M'$, $M'$ is locally of the form $M \times \mathbb{R}$ with the stable framing given by $\psi^{\mathrm{fr}} \oplus \mathrm{id}_{\mathbb{R}} : \mathbb{R}^k \oplus TM \oplus \mathbb{R} \rightarrow \mathbb{R}^{k+n+q+1}$ and the map is locally $f' = f \times \mathrm{id}_{\mathbb{R}} : M \times \mathbb{R} \rightarrow X \times W_q \times \mathbb{R}$. Then the isomorphisms $\psi(M', g', f', d)$ and $\psi(M, g, f, d)(1)$ are homotopic. +---PAGE_BREAK--- + +(3) If we change $P$ to $P \times_{s_d} G_{d+1}$ and $\mathcal{V}_d$ to $\mathcal{V}_{d+1} = \mathbb{R} \oplus \mathcal{V}_d$, the isomorphisms $\psi(M, g, f, d+1)$ and $\psi(M, g, f, d)(1)$ are homotopic. + +(4) The above niceness condition and the geometric Pontryagin-Thom construction can be extended to bordism data (Definition 3.13) in the obvious way. If we have a bordism data $(W, g_W, t_W, f_W)$ for $(M, g_M, t_M, f_M)$ with the niceness condition, $\psi(W, g_W, t_W, f_W)$ and $\psi(M, g_M, t_M, f_M)$ are related in such a way that $\text{PT}_{\nabla,\tau}(W, g_W, t_W, f_W)$ is a bordism data for $\text{PT}_{\nabla,\tau}(M, g_M, t_M, f_M)$. + +*Proof*. For (1), one can check that + +$$ \psi(M,g,f,d)(1) = (r_{(1)} \oplus \text{id}_{TM_f}) \circ \psi(M,g(1),f,d) $$ + +where $r_{(1)} : \mathbb{R}^{d+q+k+1} \to \mathbb{R}^{d+q+k+1}$ is defined by + +$$ r_{(1)} = \left[ \begin{array}{c|c|c} 0 & 1 & 0 \\ \hline -\text{id}_{\mathbb{R}^{d+q}} & 0 & 0 \\ 0 & 0 & \text{id}_{\mathbb{R}^k} \end{array} \right] \in \text{SO}(d+q+k+1). $$ + +For (2), one can check that + +$$ \psi(M,g,f,d)(1) = (r_{(2)} \oplus \text{id}_{TM_f}) \circ \psi(M',g',f',d) \circ (s_{(2)} \oplus \text{id}_{(f|_{M_f})^*\mathcal{V}_d}) $$ + +where $s_{(2)} : \mathbb{R}^{n+q+k+1} \to \mathbb{R}^{n+q+k+1}$ is defined by + +$$ s_{(2)} = \left[ \frac{0}{(-1)^{n+k+q}} \middle| \frac{\text{id}_{\mathbb{R}^{n+k+q}}}{0} \right] \in \text{SO}(n+q+k+1). $$ + +and $r_{(2)} : \mathbb{R}^{d+q+k+1} \to \mathbb{R}^{d+q+k+1}$ is defined by + +$$ r_{(2)} = \left[ \begin{array}{c|c|c|c} 0 & 0 & (-1)^q & 0 \\ \hline -\text{id}_{\mathbb{R}^d} & 0 & 0 & 0 \\ 0 & \text{id}_{\mathbb{R}^q} & 0 & 0 \\ 0 & 0 & 0 & \text{id}_{\mathbb{R}^k} \end{array} \right] \in \text{SO}(d+q+k+1). $$ + +For (3), one can check that + +$$ \psi(M,g,f,d)(1) = \psi(M,g,f,d+1) \circ (s_{(3)} \oplus \text{id}_{(f|_{M_f})^*\mathcal{V}_d}) $$ + +where $s_{(3)} : \mathbb{R}^{n+q+k+1} \to \mathbb{R}^{n+q+k+1}$ is defined by + +$$ s_{(3)} = \left[ \frac{0}{(-1)^{n+q+k}} \middle| \frac{\text{id}_{\mathbb{R}^{n+q+k}}}{0} \right] \in \text{SO}(n+q+k+1). $$ + +The point is that $r_{(1)}$ and $r_{(2)}$ are elements of $\text{SO}(d+q+k+1)$, and $s_{(2)}$ and $s_{(3)}$ are elements of $\text{SO}(n+q+k+1)$, so they can be continuously deformed to the identity maps. For this purpose we needed the sign factors $(-1)^{k+n}$ and $(-1)^{q+k+n}$ in (3.19). + +For (4), recall that the bordism relation consists of identifying the collar of $W$ with $(-\epsilon, 0] \times M$, and identifying $g_W$ with $(g_M)_{(-\epsilon,0]}$ (see (3.3)). Thus, if we use the coordinate $(z, \xi) \in \mathbb{R} \oplus TM \simeq TW$ on the collar, in the geometric Pontryagin-Thom construction for $(W, g_W, t_W, f_W)$, the isomorphism +---PAGE_BREAK--- + +corresponding to (3.20) becomes + +$$ +\begin{equation} \tag{3.25} +\begin{split} +\psi(W, g_W, f_W, d) : \underline{\mathbb{R}}^{(n+1)+q+(k-1)} &\oplus (f|_{M_f})^* \mathcal{V}_d \approx \underline{\mathbb{R}}^{d+q+k-1} \oplus \underline{\mathbb{R}} \oplus T M_f \\ +&\quad (\mathcal{A}(x', z, \xi, w), v) \mapsto (r\mathcal{B}(v, w), x', z, \xi) +\end{split} +\end{equation} +$$ + +where now $x' \in \mathbb{R}^{k-1}$. Thus identifying the collar of $W_f$ with $(-\epsilon, 0] \times M_f$, +the isomorphisms (3.20) and (3.25) coincide. This means that $\mathrm{PT}_{\nabla,\tau}(W,g_W, t_W, f_W)$ +is a bordism data for $\mathrm{PT}_{\nabla,\tau}(M,g_M, t_M, f_M)$ and completes the proof. $\square$ + +Finally, we observe the compatibility of the boundary operator $\partial: C^G_{\Delta,n}(X,A) \to C^G_{\Delta,n-1}(A)$ in (3.11) with the geometric Pontryagin-Thom construction. Given a nice $(M,g,t,f)$ as above, its boundary $\partial(M,g,t,f) = (\partial M, \partial g, \partial t, f|_{\partial M})$ is a geometric singular stable tangential fr-cycle $(M,g,t,f)$ of dimension $(n+q-1)$ over $A \times \text{Thom}_\infty(W_q)$, and it is also nice with the induced tubular neighborhood structure $\pi_f|_{\partial M}$. Then we have + +$$ +(3.26) \qquad \partial(\mathrm{PT}_{\nabla,\tau}(M,g,t,f)) = \mathrm{PT}_{\nabla,\tau}(\partial(M,g,t,f)). +$$ + +This is checked in the same way as the proof of Lemma 3.24 (4). + +4. PHYSICALLY MOTIVATED MODELS FOR THE ANDERSON DUALS OF G-BORDISMS + +Let $G = \{G_d, s_d, \rho_d\}_{d \in \mathbb{Z}_{\ge 0}}$ be as in Subsection 2.3. In this section, we give new models $(I\Omega_{\mathrm{dR}}^G)^*$ and $(I\Omega_{\mathrm{sing}}^G)^*$ of the generalized cohomology theory $(I\Omega^G)^*$, the Anderson dual to $G$-bordism theory. + +Let us define a $G_d$-module $\mathbb{R}_{G_d}$ with the underlying vector space $\mathbb{R}$, and +the $G_d$-module structure given by the multiplication via $G_d \xrightarrow{\rho_d} O(d, \mathbb{R}) \xrightarrow{\det} \{\pm 1\}$. + +**Lemma 4.1.** We have + +$$ +(I\Omega^G)^*(pt) \otimes \mathbb{R} = \lim_{\leftarrow d} H^*(G_d; \mathbb{R}_{G_d}). +$$ + +Here $H^*(G_d; \mathbb{R}_{G_d})$ is the group cohomology of $G_d$ with coefficient in $\mathbb{R}_{G_d}$. + +Proof. By (2.33), we have + +$$ +\begin{align*} +(I\Omega^G)^n(\text{pt}) \otimes \mathbb{R} &= \operatorname{Hom}(\Omega^n_G(\text{pt}), \mathbb{Z}) \otimes \mathbb{R} = \operatorname{Hom}\left(\lim_{\underset{q}{\longrightarrow}} \pi_{n+q}((MTG)_q), \mathbb{R}\right) \\ +&= \lim_{\underset{q}{\longleftarrow}} H^{n+q}((MTG)_q; \mathbb{R}) = \lim_{\underset{q}{\longleftarrow}} \lim_{\underset{d}{\longleftarrow}} H^{n+q}((\Sigma^d MTG_d)_q; \mathbb{R}). +\end{align*} +$$ + +Recall that the Madsen-Tillmann spectrum *MTG* consists of the Thom spaces of the normal bundles *Q**q* of the universal bundles over approximations *X**d*(*d* + *q*) of *BG**d* (Definition 2.23). We see that the orientation bundle *Ori*(*Q**q* → *X**d*(*d* + *q*)) over *X**d*(*d* + *q*) is the pullback of the bundle *EG**d*×*G**d* R*G**d* over *BG**d*. By the Thom isomorphism theorem, we have + +$$ +\begin{align*} +\lim_{\leftarrow q} H^{n+q}((\Sigma^d MTG_d)_q; \mathbb{R}) &\approx \lim_{\leftarrow q} H^n(\mathcal{X}_d(d+q); EG_d|\mathcal{X}_d(d+q) \times_{G_d} R_{G_d}) \\ +&\approx H^n(BG_d; EG_d \times_{G_d} R_{G_d}) := H^n(G_d; R_{G_d}). +\end{align*} +$$ +---PAGE_BREAK--- + +By the commutative diagram + +$$ +\begin{tikzcd} +& & \mathcal{X}_d(d+q) \arrow[r] & \mathcal{X}_d(d+q+1) \\ +& & \mathcal{X}_{d+1}(d+q+1) \arrow[u] \arrow[r, " \circ ", " \sim "'] & \mathcal{X}_{d+1}(d+q+2) \\ +& & & \\ +& & \mathcal{X}_{d+q}(d+q) \arrow[r] & \mathcal{X}_{d+q+1}(d+q) +\end{tikzcd} +$$ + +the two limits $\lim_{\leftarrow q}$ and $\lim_{\rightarrow d}$ commute. So we get the result. $\square$ + +**Lemma 4.2.** *Fix a nonnegative integer $d$. For each integer $n$, we have* + +$$ +H^{2n}(G_d; \mathbb{R}G_d) \simeq (\mathrm{Sym}^n g_d^* \otimes_R \mathbb{R}G_d)^{G_d}, +$$ + +$$ +H^{2n+1}(G_d; \mathbb{R}G_d) = 0. +$$ + +Here $\mathfrak{g}_d$ is the Lie algebra of $G_d$ and $\mathfrak{g}_d^*$ is its dual. The notation $(-)^{G_d}$ means the $G_d$-invariant part, where $G_d$ acts on $\mathfrak{g}_d$ by the adjoint. + +Proof. Let $K_d$ be the kernel of the homomorphism $\det \circ \rho_d: G_d \to \{\pm 1\}$. +In the case where $G_d = K_d$, the $G_d$-module $\mathbb{R}G_d$ is trivial $\mathbb{R}$, and the desired results follow from the Chern-Weil isomorphism for $G_d$. In the case where $G_d \neq K_d$, apply the Hochschild-Serre spectral sequence for the group extension + +$$ +1 \to K_d \to G_d \to \{\pm 1\} \to 1 +$$ + +and the $G_d$-module $\mathbb{R}G_d$. We get the isomorphism $H^*(G_d; \mathbb{R}G_d) \simeq (H^*(K_d; \mathbb{R}) \otimes_{\mathbb{R}} \mathbb{R}G_d)^{\{\pm 1\}} = (H^*(K_d; \mathbb{R}) \otimes_{\mathbb{R}} \mathbb{R}G_d)^{G_d}$. Using the Chern-Weil isomorphism for $K_d$, we get the result. $\square$ + +By this reason, we denote (see (2.56)) + +$$ +(4.3) \quad V_{I\Omega^G}^\bullet := \lim_{\leftarrow d} H^\bullet(BG_d; \mathbb{R}G_d) = \lim_{\leftarrow d} (\text{Sym}^{\bullet/2} g_d^* \otimes_R \mathbb{R}G_d)^{G_d}. +$$ + +In the case where $G$ is oriented, i.e., the image of $\rho_d$ lies in $SO(d, \mathbb{R})$ for +each $d$, the $G_d$-module $\mathbb{R}G_d$ is trivial and $V_{I\Omega^G}^\bullet$ is the projective limit of +invariant polynomials on $\mathfrak{g}_d$. In general cases, $V_{I\Omega^G}^\bullet$ is the projective limit +of polynomials on $\mathfrak{g}_d$ which change the sign according to the action of $G_d$. + +4.1. **The models.** + +4.1.1. The differential models. In this subsubsection, we define the differen- +tial model $(I\Omega_{d\mathbb{R}}^G)^*$ on the category of manifolds. We first construct a model +$(I\widehat{\Omega}_{d\mathbb{R}}^G)^*$ of a differential extension of $(I\Omega^G)^*$, and $(I\Omega_{d\mathbb{R}}^G)^*$ is constructed as +its quotient. The definition uses a variant of Chern-Weil construction, as we +now explain. + +**Definition 4.4.** (1) Let $W$ be a manifold and $(P, \nabla)$ be a principal $G_d$-bundle with connection on it. Let $n \in 2\mathbb{Z}_{\ge 0}$ and $\phi \in V_{I\Omega^G}^n$. Apply the forgetful map + +$$ +(4.5) \qquad V_{I\Omega^G}^{\bullet} \to V_{I\Omega^{G_d}}^{\bullet} = (\mathrm{Sym}^{\bullet/2} g_d^* \otimes_R \mathbb{R} G_d)^{G_d}. +$$ + +to $\phi$ and denote by $\phi_d$ the resulting element. We set + +$$ +(4.6) \quad \mathrm{cw}\nabla(\phi) = \mathrm{cw}\nabla(\phi_d) := \phi_d(F\nabla) \in \Omega_{\mathrm{clo}}^n(W; P \times_{G_d} \mathbb{R}G_d), +$$ +---PAGE_BREAK--- + +where $F_{\nabla} \in \Omega_{\text{clo}}^2(W; P \times_{\text{ad}} \mathfrak{g}_d)$ is the curvature form for $(P, \nabla)$, +and we used the identification $\mathbb{R}_{G_d} = \mathbb{R}^*_{G_d}$ of $G_d$-modules. This +construction gives a homomorphism $\text{cw}_{\nabla}: V_{I^{\Omega_G}}^n \to \Omega_{\text{clo}}^n(W; P \times_{G_d} \mathbb{R}_{G_d})$. Extending it $\Omega^*(W)$-linearly, we get a homomorphism of $\mathbb{Z}$-graded real vector spaces, + +$$ (4.7) \qquad \text{cw}_{\nabla} : \Omega^*(W; V_{I^{\Omega_G}}^{\bullet}) \to \Omega^*(W; P \times_{G_d} \mathbb{R}_{G_d}), $$ + +which restricts to a homomorphism $\text{cw}_{\nabla}: \Omega_{\text{clo}}^*(W; V_{I^{\Omega_G}}^{\bullet}) \to \Omega_{\text{clo}}^*(W; P \times_{G_d} \mathbb{R}_{G_d})$. Explicitly, given $\omega \in \Omega^*(W; V_{I^{\Omega_G}}^{\bullet})$, we apply (4.5) on the co-efficient and denote the resulting element by $\omega_d$. Write it as a finite sum $\omega_d = \sum_i \xi_i \otimes \phi_i$ with $\xi_i \in \Omega^*(X, A)$ and $\phi_i \in V_{I^{\Omega^{G_d}}}^{\bullet}$. This is possible because $V_{I^{\Omega^{G_d}}}^{\bullet}$ is finite dimensional. Then set + +$$ (4.8) \qquad \text{cw}_{\nabla}(\omega) := \sum_i \xi_i \wedge \text{cw}_{\nabla}(\phi_i) $$ + +(2) If $W$ is a manifold with a geometric stable tangential $G$-structure $g = +(d, P, \nabla, \psi)$, the isomorphism $\psi$ induces the isomorphism $\psi: P \times_{G_d} \mathbb{R}_{G_d} \simeq \text{Ori}(W)$. Composing the homomorphism (4.7) with this $\psi$ on the coefficient, we define a homomorphism of $\mathbb{Z}$-graded real vector spaces, + +$$ (4.9) \qquad \text{cw}_g := \psi \circ \text{cw}_{\nabla} : \Omega^*(W; V_{I^{\Omega_G}}^{\bullet}) \to \Omega^*(W; \text{Ori}(W)), $$ + +which restricts to the homomorphism $\text{cw}_g : \Omega_{\text{clo}}^*(W; V_{I^{\Omega_G}}^{\bullet}) \to \Omega_{\text{clo}}^*(W; \text{Ori}(W))$. + +**Remark 4.10.** In Definition 4.4 (2), the homomorphism $\text{cw}_g$ actually depends on $\psi$ in $g = (d, P, \nabla, \psi)$ only through its *homotopy class*. This is because we only used $\psi$ to construct the isomorphism $\psi: P \times_{G_d} \mathbb{R}_{G_d} \simeq \text{Ori}(W)$, and it does not change when we replace $\psi$ to a homotopic one. + +Suppose we are given a pair of manifolds $(X, A)$ and a form $\omega \in \Omega^*(X, A; V_{I^{\Omega_G}}^\bullet)$. If $W$ is a manifold equipped with a geometric stable tangential $G$-structure $g$, given a map $f \in C^\infty((W, \emptyset), (X, A))$, the homomorphism (4.9) gives + +$$ \text{cw}_g(f^*\omega) \in \Omega^*(W, f^{-1}(A); \text{Ori}(W)). $$ + +If $\omega \in \Omega_{\text{clo}}^n(X, A; V_{I^{\Omega_G}}^\bullet)$, the above construction induces a homomorphism + +$$ \text{cw}(\omega) : \Omega_n^G(X, A) \to \mathbb{R}, \quad [(W, g, f)] \mapsto \int_W \text{cw}_g(f^*\omega). $$ + +This gives the isomorphism $H^n(X, A; V_{I^{\Omega_G}}^\bullet) \cong \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R})$ which +follows from the Hurewicz isomorphism and Künneth formula + +$$ \operatorname{Hom}(\pi_n((X/A, {}_*)) \wedge MTG, \mathbb{R}) \simeq H^n((X/A, {}_*) \wedge MTG, \mathbb{R}) \simeq H^n(X, A; H^\bullet(MTG, \mathbb{R})). $$ + +**Definition 4.11** (($I\widehat{\Omega}_{\mathrm{dR}}^G$)* and $(I\Omega_{\mathrm{dR}}^G)^*$). Let $(X, A)$ be a pair of manifolds and $n \in \mathbb{Z}_{\ge 0}$. + +(1) Define $(I\widehat{\Omega}_{\mathrm{dR}}^G)^n(X, A)$ to be an abelian group consisting of pairs $(\omega, h)$, such that + +(a) $\omega$ is a closed $n$-form $\omega \in \Omega_{\mathrm{clo}}^n(X, A; V_{I^{\Omega_G}}^\bullet)$. + +(b) $h$ is a group homomorphism $h: C_{\infty, n-1}^G(X, A) \to \mathbb{R}/\mathbb{Z}$. +---PAGE_BREAK--- + +(c) $\omega$ and $h$ satisfy the following compatibility condition. Assume that we are given a bordism data $(W, g_W, f_W)$ of a geometric smooth stable tangential $G$-cycle $(M, g_M, f_M)$ of dimension $(n-1)$ over $(X, A)$ (Definition 3.8). Then we have + +$$ (4.12) \qquad h([M, g_M, f_M]) = \int_W \mathrm{cw}_{g_W}(f_W^*\omega) \pmod{\mathbb{Z}}. $$ + +Abelian group structure is defined in the obvious way. + +(2) We define a homomorphism of abelian groups, + +$$ \begin{align*} +(4.13) \quad & a: \Omega^{n-1}(X, A; V_{I\Omega^G}^\bullet)/\mathrm{Im}(d) \to (\widehat{I\Omega_{dR}^G})^n(X, A) \\ +& \phantom{a:} \alpha \mapsto (d\alpha, \mathrm{cw}(\alpha)). +\end{align*} $$ + +Here the homomorphism $\mathrm{cw}(\alpha): C_{\infty,n-1}^G(X,A) \to \mathbb{R}/\mathbb{Z}$ is defined by + +$$ (4.14) \qquad \mathrm{cw}(\alpha)([M, g, f]) := \int_M \mathrm{cw}_g(f^*\alpha) \pmod{\mathbb{Z}}. $$ + +We set + +$$ (I\Omega_{dR}^G)^n(X, A) := (\widehat{I\Omega_{dR}^G})^n(X, A)/\mathrm{Im}(a). $$ + +For $n \in \mathbb{Z}_{<0}$ we set $(\widehat{I\Omega_{dR}^G})^n(X, A) = 0$ and $(I\Omega_{dR}^G)^n(X, A) = 0$. + +For a smooth map $\phi \in C^\infty((X, A), (Y, B))$ between two pairs of manifolds, by the pullback we get the homomorphisms $\phi^*: (\widehat{I\Omega_{dR}^G})^n(Y, B) \to (\widehat{I\Omega_{dR}^G})^n(X, A)$ and $\phi^*: (I\Omega_{dR}^G)^n(Y, B) \to (I\Omega_{dR}^G)^n(X, A)$. Thus we get covariant functors, + +$$ (\widehat{I\Omega_{dR}^G})^*, (\widehat{I\Omega_{dR}^G})^* : \mathrm{MfdPair}^{\mathrm{op}} \to \mathrm{Ab}^\mathbb{Z}. $$ + +In the case where $G_d = 1$ for all $d$, i.e., the case of the stably framed bordism theory $\Omega^{\mathrm{fr}}$, the corresponding Madsen-Tillmann spectrum is the sphere spectrum and $(I\Omega^{\mathrm{fr}})^* = I\mathbb{Z}^*$. So in this case, we also denote $(\widehat{I\mathbb{Z}_{dR}})^* := (\widehat{I\Omega_{dR}^{\mathrm{fr}}})^*$ and $(I\mathbb{Z}_{dR})^* := (\widehat{I\Omega_{dR}^{\mathrm{fr}}})^*$. + +We sometimes say that "evaluate $(\omega, h)$ on $[M, g, f]$" to just mean getting the value $h([M, g, f])$. + +The functor $(\widehat{I\Omega_{dR}^G})^*$ is equipped with the following structure maps. In Proposition 4.122, we will see that it makes $(\widehat{I\Omega_{dR}^G})^*$ into a differential extension of $(I\Omega^G)^*$. + +**Definition 4.15** (Structure maps for $(\widehat{I\Omega_{dR}^G})^*$ and $(I\Omega_{dR}^G)^*$). We define the following maps natural in $(X, A)$. The well-definedness is easy by Definition 4.11. + +• We denote the quotient map by + +$$ I: (\widehat{I\Omega_{dR}^G})^*(X, A) \to (I\Omega_{dR}^G)^*(X, A). $$ + +• We define + +$$ R: (\widehat{I\Omega_{dR}^G})^*(X, A) \to \Omega_{\mathrm{clo}}^*(X, A; V_{I\Omega^G}^\bullet), \quad (\omega, h) \mapsto \omega. $$ + +• We define + +$$ \mathrm{ch}: (\widehat{I\Omega_{dR}^G})^*(X, A) \to H^n(X, A; V_{I\Omega^G}^\bullet) (\simeq \mathrm{Hom}(\Omega_*^G(X, A), \mathbb{R})), \quad I((\omega, h)) \mapsto [\omega] (\mapsto \mathrm{cw}(\omega)). $$ +---PAGE_BREAK--- + +• We define + +$$p: \operatorname{Hom}(\Omega_{*-1}^G(X, A), \mathbb{R}/\mathbb{Z}) \to (\widetilde{I\Omega}_{\mathrm{dR}}^G)^*(X, A), \quad h \mapsto I((0, h)),$$ + +where we regard $h \in \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}/\mathbb{Z})$ as a group homomorphism $h: C_{\infty,n-1}^G(X, A) \to \mathbb{R}/\mathbb{Z}$. + +In the physical interpretation of the group $(\widetilde{I\Omega}_{\mathrm{dR}}^G)^n(X)$ explained in Subsection 1.2, an invertible $(n-1)$-dimensional QFT on manifolds equipped with geometric stable tangential $G$-structures and maps to $X$ gives an element $(\omega, h) \in (\widetilde{I\Omega}_{\mathrm{dR}}^G)^n(X)$. In this picture, the value $\exp(2\pi\sqrt{-1}h([M, g, f]))$ corresponds to the value of partition function applied to $[M, g, f] \in C_{\infty,n-1}^G(X)$. + +Now we observe an easy property of $h$. Recall that a geometric stable tangential $G$-structure is a quadruple $g = (d, P, \nabla, \psi)$. We can show that the value of $h$ is unchanged if we change it to $g' = (d, P, \nabla, \psi')$, where the principal bundle and the connection is fixed, but $\psi$ is replaced with another $\psi'$ which is homotopic to $\psi$, as follows. + +**Lemma 4.16.** Let $(X, A)$ be a pair of manifolds, $n$ be a positive integer, and assume we are given an element $(\omega, h) \in (\widetilde{I\Omega}_{\mathrm{dR}}^G)^n(X, A)$. Let $(M, g, f)$ be a geometric smooth stable tangential $G$-cycle of $(n-1)$-dimension over $(X, A)$ with $g = (d, P, \nabla, \psi)$. Assume we change the isomorphism $\psi: P \times_{\rho_d} \mathbb{R}^d \simeq \mathbb{R}^{d-n} \oplus TM$ to a homotopic one, $\psi'$, and denote the resulting geometric stable tangential $G$-structure on $M$ by $g' = (d, P, \nabla, \psi')$. Then we have + +$$h([M, g, f]) = h([M, g', f]).$$ + +*Proof.* By stabilizing $g$ and $g'$ if necessary, we may assume $d > n$. Using a homotopy $\tilde{\psi}$ connecting $\psi$ and $\psi'$, form a bordism data $([0,1] \times M, \tilde{g}, f \circ \text{pr}_M)$ for $(M,g,f) \sqcup (M,g'_{\text{op}},f)$, where $\tilde{g} := (d, \text{pr}_M^*P, \text{pr}_M^*\nabla, \tilde{\psi})$. The point is that the $G_d$-connection is constant in $[0,1]$-direction. Then we apply the compatibility condition for $(\omega,h)$ in Definition 4.11 (1) (c) for this bordism data. Since we have $\int_{[0,1]\times M} \text{cw}_{\tilde{g}}((f \circ \text{pr}_M)^*\omega) = \int_{[0,1]\times M} \text{pr}_M^* \text{cw}_g(f^*\omega) = 0$, we get the result. $\square$ + +Another important consequence of the compatibility condition of $\omega$ and $h$ in Definition 4.11 is the following integrality condition. + +**Lemma 4.17** (The integrality condition). *We have* + +$$ (4.18) \qquad \operatorname{Im}(\operatorname{ch}) \subset \operatorname{Hom}(\Omega_*^G(X, A), \mathbb{Z}), $$ + +i.e., any element $(\omega, h) \in (\widetilde{I\Omega}_{\mathrm{dR}}^G)^*(X, A)$ induces a $\mathbb{Z}$-valued homomorphism $\text{cw}(\omega): \Omega_*^G(X, A) \to \mathbb{Z}$. + +*Remark 4.19.* The inclusion (4.18) is actually an equality by Proposition 4.20. + +*Proof.* Take any element $[W,g_W,f_W] \in C_{\infty,n}^G(X,A)$. It is enough to prove that $\int_W c w_{g_W}(f_W^*\omega) \in \mathbb{Z}$. But this follows from the condition (c) in Definition 4.11 (1) applied to $M = \emptyset$. $\square$ +---PAGE_BREAK--- + +The proof that the functor $(I\Omega_{\mathrm{DR}}^G)^*$ is actually isomorphic to $(I\Omega^G)^*$ is very nontrivial, and actually it is the main result of this paper. However, the fact that it fits into the same exact sequence as in (2.33), which is an important property of the Anderson dual, is easier, as follows. + +**Proposition 4.20.** + +For a pair of manifolds $(X, A)$ and $n \in \mathbb{Z}_{\ge 0}$, the following sequence is exact. + +$$ +\begin{align*} +(4.21) \quad \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}) &\to \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}/\mathbb{Z}) \xrightarrow{p} (I\Omega_{\mathrm{DR}}^G)^n(X, A) \\ +&\xrightarrow{\mathrm{ch}} \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}) \to \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}/\mathbb{Z}). +\end{align*} +$$ + +To prove Proposition 4.20, we need the following lemma. + +**Lemma 4.22.** Let $(X, A)$ be a pair of manifolds and $n \in \mathbb{Z}_{\ge 0}$. Let $\omega \in \Omega_{\mathrm{clo}}^n(X, A; V_{I\Omega^G}^\bullet)$ be a closed form which induces an integer-valued homomorphism $\mathrm{cw}(\omega): \Omega_n^G(X, A) \to \mathbb{Z}$. + +Let $(M, g_M, f_M)$ be a geometric smooth stable tangential $G$-cycle over $(X, A)$ of dimension $(n-1)$. Assume that we are given two bordism data $(W, g_W, f_W)$ and $(W', g_{W'}$, $f_{W'}$) for $(M, g_M, f_M)$. Then we have + +$$ +(4.23) \quad \int_W \mathrm{cw}_{g_W}(f_W^*\omega) = \int_{W'} \mathrm{cw}_{g_{W'}}(f_{W'}^*\omega) \pmod{\mathbb{Z}}. +$$ + +Proof of Lemma 4.22. First we remark that, the proof is easy if $G$ is such that “reversing the stable tangential $G$-structure” $g_{W'} \mapsto \bar{g}_{W'}$ makes sense so that $(W', \bar{g}_{W'})$, $f_{W'}$ is a bordism data for $(M, (g_M)_{\mathrm{op}}$, $f_M$). In this case, we can glue two bordism data together to get $(W \cup_M W'$, $g_W \cup \bar{g}_{W'}$, $f_W \cup f_{W'}$) which defines an element in $C_{\infty,n}^G(X,A)$. If we apply the integrality condition for $\omega$ (Lemma 4.17), we get the result. Examples of such $G$ includes O, SO, and Spin. However, for example if $G=\mathrm{fr}$ such a reversing does not make sense. In the following we consider a gluing procedure which works for any $G$. + +Let $g_M = (d, P_M, \nabla_M, \psi_M)$, $g_W = (d, P_W, \nabla_W, \psi_W)$ and $g_{W'} = (d, P_{W'}, \nabla_{W'}, \psi_{W'})$. + +By Definition 3.5, we have the following data. + +• Isomorphisms $\varphi: \partial_0 W \simeq M$ and $\varphi': \partial_0 W' \simeq M$ of $\langle 1 \rangle$-manifolds with collar structures. + +• Open neighborhoods $U \subset W$ and $U' \subset W'$ of $\partial_0 W$ and $\partial_0 W'$, respectively. + +• A positive number $\epsilon > 0$. + +• Diffeomorphisms $\phi: (-\epsilon, 0] \times M \simeq U$ and $\phi': (-\epsilon, 0] \times M \simeq U'$ which extend $\phi^{-1}$ and $\phi'^{-1}$, respectively. + +• Isomorphisms of geometric stable tangential $G$-structures $\xi: (g_M)(-\epsilon, 0] \simeq g_W|_U$ and $\xi': (g_M)(-\epsilon, 0] \simeq g_{W'}|_U$ which cover $\phi$ and $\phi'$, respectively. +This means that we have the following commutative diagrams, + +$$ +(4.24) \quad +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] + (\text{pr}_M^*P_M, \text{pr}_M^*\nabla_M) \arrow[r, "\xi", "u"] & (P_W, \nabla_W)|_U \arrow[d] \\ + (-\epsilon, 0] \times M \arrow[u,r] \arrow[d, "u"] & U \arrow[u,r] \\ + (-\epsilon, 0] \times M \arrow[u,l] & U +\end{tikzcd} +\quad +\begin{aligned} +& \text{pr}_M^* P_M \times_{\rho_d} \mathbb{R}^d \xrightarrow{\psi_M} \mathbb{R}^{d-n} \oplus \mathbb{R} \oplus \text{pr}_M^* TM \\ +& \cong \left|\xi\right| \circ \left|\psi_W\right| \\ +& P_W|_U \times_{\rho_d} \mathbb{R}^d \xrightarrow{\cong} \mathbb{R}^{d-n} \oplus TU +\end{aligned} +$$ +---PAGE_BREAK--- + +and also the corresponding commutative diagrams for $\xi'$. + +The strategy of the proof is to deform the geometric stable tangential $G$-structures $g_M$, $g_W$ and $g_{W'}$ so that “gluing along $M$” makes sense, and use the integrality condition of $\omega$. In order for that, recall the stabilization of the geometric stable tangential $G$-structures defined in Definition 3.6. We are actually going to deform $g_M(1)$, $g_W(1)$ and $g_{W'}(1)$. We write $g_M(1) = (d+1, P_M(1) = P_M \times_{s_d} G_{d+1}, \nabla_M(1), \psi_M(1))$ and similarly for $g_W(1)$ and $g_{W'}(1)$. It is obvious that $(W, g_W(1), f_W)$ and $(W', g_{W'}(1), f_{W'})$ are bordism data of $(M, g_M(1), f)$, with the isomorphism naturally induced by $\xi$ and $\xi'$, which is denoted by $\xi(1): (g_M(1))_{(-\epsilon,0]} \simeq g_W(1)|_U$ and $\xi'(1): (g_M(1))_{(-\epsilon,0]} \simeq g_{W'}(1)|_{U'}$. The commutative diagram corresponding to (4.24) becomes + +$$ (4.25) \quad \begin{tikzcd}[column sep=2.8em, row sep=2.8em] + (\mathrm{pr}_M^* P_M(1), \mathrm{pr}_M^*\nabla_M(1)) & \xrightarrow{\xi(1)} & (P_W(1), \nabla_W(1))|_U, \\ + (-\epsilon, 0] \times M & \xrightarrow{\phi} & U + \end{tikzcd} $$ + +$$ (4.26) \quad \begin{aligned} \mathrm{pr}_M^* P_M(1) \times_{\rho_{d+1}} \mathbb{R}^{d+1} &\xrightarrow{\sim} \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus \mathbb{R} \oplus \mathrm{pr}_M^* TM \\ &\approx \left|\xi(1)\right| \circ \xrightarrow{\sim} \left|\mathrm{id}_{\mathbb{R}\oplus\mathbb{R}^{d-n}\oplus\phi}\right| \\ P_W(1)|_U \times_{\rho_{d+1}} \mathbb{R}^{d+1} &\xrightarrow{\sim} \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus TU \end{aligned} $$ + +First we pick a smooth function $c: (-\epsilon, 0] \to \text{SO}(d-n+2, \mathbb{R})$ satisfying + +$$ c(t) = \begin{cases} \text{id}_{\mathbb{R}^{d-n+2}} & t \in (-\epsilon, -2\epsilon/3] \\ \tau_{1,d-n+2} & t \in [-\epsilon/3, 0]. \end{cases} $$ + +Here $\tau_{1,d-n+2} \in \text{SO}(d-n+2, \mathbb{R})$ is the element defined as + +$$ \tau_{1,d-n+2} := \left[ \begin{array}{c|c|c} 0 & 0 & 1 \\ \hline 0 & \text{id}_{\mathbb{R}^{d-n}} & 0 \\ -1 & 0 & 0 \end{array} \right] \in \text{SO}(d-n+2, \mathbb{R}). $$ + +We can interpolate between $-2\epsilon/3$ and $-\epsilon/3$ by rotating the first and the last coordinate of $\mathbb{R}^{d-n+2}$. Using this, we define the following automorphisms on the stable tangent bundles. + +• The automorphism $c_M$ on the vector bundle $\mathbb{R}^{d-n+2} \oplus TM$ over $M$ is defined as + +$$ c_M := \tau_{1,d-n+2} \oplus \mathrm{id}_{TM} : \mathbb{R}^{d-n+2} \oplus TM \to \mathbb{R}^{d-n+2} \oplus TM. $$ + +• The automorphism $c_W$ on the vector bundle $\mathbb{R}^{d-n+1} \oplus TW$ over $W$ is defined as follows. On $U$, we define it to be the following composition, + +$$ c_W|_U := \left\{ \begin{array}{r@{\,}c@{\,}l@{\quad}l} \mathbb{R}^{d-n+1} & TU & \xrightarrow{\mathrm{id}_{\mathbb{R}^{d-n+1}} \oplus \phi^{-1}} \mathbb{R}^{d-n+2} \oplus \mathrm{pr}_M^* TM \\ & & \xrightarrow{\mathrm{id}_{\mathrm{pr}_M^* TM}} \mathbb{R}^{d-n+2} \oplus \mathrm{pr}_M^* TM \\[2ex] & & \xrightarrow{\mathrm{id}_{\mathbb{R}^{d-n+1}} \oplus \phi} \mathbb{R}^{d-n+1} \oplus TU \end{array} \right\}. $$ +---PAGE_BREAK--- + +and on $W\backslash U$, we set $c_W|_{W\backslash U} := \text{id}$. This gives a well-defined smooth automorphism. + +* We also define the automorphism $c_{W'}$ on the vector bundle $\mathbb{R}^{d-n+1} \oplus TW'$ over $W'$ analogously to $c_W$. + +* The automorphism $r_{W'}$ on the vector bundle $\mathbb{R}^{d-n+1} \oplus TW'$ over $W'$ is defined as + +$$r_{W'} := (-\mathrm{id}_{\mathbb{R}}) \oplus \mathrm{id}_{\mathbb{R}^{d-n}} \oplus \mathrm{id}_{TW'} : \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus TW' \to \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus TW'.$$ + +Now we introduce new geometric stable tangential $G$-structures $g_M(1)_{\text{def}}$ on $M$, $g_W(1)_{\text{def}}$ on $W$ and $g_{W'}(1)_{\text{def,rev}}$ on $W'$, as follows. Recall that we denote $g_M(1) = (d+1, P_M(1), \nabla_M(1), \psi_M(1))$, $g_W(1) = (d+1, P_W(1), \nabla_W(1), \psi_W(1))$ and $g_{W'}(1) = (d+1, P_{W'}(1), \nabla_{W'}(1), \psi_{W'}(1))$. + +* We set $g_M(1)_{\text{def}} := (d+1, P_M(1), \nabla_M(1), c_M \circ \psi_M(1))$. + +* We set $g_W(1)_{\text{def}} := (d+1, P_W(1), \nabla_W(1), c_W \circ \psi_W(1))$. + +* We set $g_{W'}(1)_{\text{def,rev}} := (d+1, P_{W'}(1), \nabla_{W'}(1), c_{W'} \circ r_{W'} \circ \psi_{W'}(1))$. + +$g_W(1)_{\text{def}}$ plays the role of "g$_W(1)$ deformed on the collar", and $g_{W'}(1)_{\text{def,rev}}$ plays the role of "g$_{W'}(1)$ deformed on the collar and reversed". The key properties of these geometric stable tangential $G$-structures are the following. + +**Claim 4.27.** (1) We have $cw_{g_W(1)_{\text{def}}} (f_W^*\omega) = cw_{g_W}(f_W^*\omega)$. + +(2) We have $cw_{g_{W'}(1)_{\text{def,rev}}} (f_{W'}^*\omega) = -cw_{g_{W'}} (f_{W'}^*\omega)$. + +Let us denote $U_{\epsilon/3} := \phi((-\epsilon/3, 0] \times M)$ and $U'_{\epsilon/3} := \phi((-\epsilon/3, 0] \times M)$. +Let $\sigma: [0, \epsilon/3) \times M \to (-\epsilon/3, 0] \times M$ be the diffeomorphism defined by +$\sigma(t, x) := (-t, x)$. In the following statement, we denote the projections +by $\text{pr}_{+,M}: [0, \epsilon/3) \times M \to M$ and $\text{pr}_{-,M}: (-\epsilon/3, 0] \times M \to M$. + +(3) The isomorphism $\xi(1)|_{(-\epsilon/3,0]\times M} : \text{pr}^*_{-,M} P_M(1) \to P_W(1)|_{U_{\epsilon/3}}$ in (4.25) +gives the isomorphism of geometric stable tangential $G$-structures + +$$\xi(1)_{\text{def}} : (g_M(1)_{\text{def}})_{(-\epsilon/3,0]} \cong g_W(1)_{\text{def}}|_{U_{\epsilon/3}}$$ + +which covers the diffeomorphism $\phi|_{(-\epsilon/3,0]\times M}: (-\epsilon/3,0] \times M \to U_{\epsilon/3}$. + +(4) Denote by $\sigma: \text{pr}^*_{+,M} P_M(1) \to \text{pr}^*_{-,M} P_M(1)$ the isomorphism induced by the relation $\text{pr}_{-,M} \circ \sigma = \text{pr}_{+,M}$. The isomorphism $\xi'(1)|_{(-\epsilon/3,0]\times M} : \text{pr}^*_{+,M} P_M(1) \to P_W(1)|_{U_{\epsilon/3}}$ gives the isomorphism of geometric stable tangential $G$-structures + +$$\xi'(1)_{\text{def,rev}} : (g_M(1)_{\text{def}})_{[0,\epsilon/3)} \simeq g_{W'}(1)_{\text{def,rev}}|_{U'_{\epsilon/3}}$$ + +which covers the diffeomorphism $\phi'|_{(-\epsilon/3,0]\times M} \circ \sigma: [0, \epsilon/3) \times M \to U'_{\epsilon/3}$. + +*Proof of Claim 4.27.* (1) and (2) easily follow by Remark 4.10 and the definitions of $g_W(1)_{\text{def}}$ and $g_{W'}(1)_{\text{def,rev}}$, since we do not change the principal bundles and connections. The minus sign in (2) is because the isomorphism $r_{W'}$ reverses the orientation of the first $\mathbb{R}$. (3) is the consequence of the +---PAGE_BREAK--- + +commutativity of the following diagram where vertical arrows cover the dif- +feomorphism $\phi|_{(-\epsilon/3,0] \times M} : (-\epsilon/3, 0] \times M \to U_{\epsilon/3}$. + +$$ +\begin{tikzcd} +& & \mathrm{pr}_{-,-,M}^* P_M(1) \times_{\rho_{d+1}} \mathbb{R}^{d+1} \arrow[r, "c_M"] & \\ +& & \xi(1)|_{(-\epsilon/3,0]\times M} \arrow[u] \arrow[r, "c_W|_{U_{\epsilon/3}}"] & \mathrm{id}_{\mathbb{R}\oplus\mathbb{R}}^{d-n} \oplus \phi \arrow[u] \\ +& & P_W(1)|_{U_{\epsilon/3}} \times_{\rho_{d+1}} \mathbb{R}^{d+1} \arrow[r, "c_W|_{U_{\epsilon/3}}", "id_{\mathbb{R}\oplus\mathbb{R}}^{d-n} \oplus \phi"] & \mathrm{id}_{\mathbb{R}\oplus\mathbb{R}}^{d-n} \oplus \phi \arrow[u] +\arrow[r, "c_M"] & \mathrm{pr}_{-,-,M}^* P_M(1) \times_{\rho_{d+1}} \mathbb{R}^{d+1} \arrow[r, "c_M"] & +\end{tikzcd} +$$ + +Finally, we prove (4). We have + +$$ +(4.28) \quad \tau_{1,d-n+2} \circ ((-\mathrm{id}_\mathbb{R}) \oplus \mathrm{id}_{\mathbb{R}^{d-n+1}}) = (\mathrm{id}_{\mathbb{R}^{d-n+1}} \oplus (-\mathrm{id}_\mathbb{R})) \circ \tau_{1,d-n+2} +$$ + +on $\mathbb{R}^{d-n+2}$. Consider the following diagram of vector bundles, where the vertical arrows cover the diffeomorphisms $[0, \epsilon/3) \times M \xrightarrow{\sigma} (-\epsilon/3, 0] \times M \xrightarrow{\phi'} [-\epsilon/3, 0] \times M$ +$U_{\epsilon/3}'$. + +$$ +(4.29) +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] + & \mathrm{pr}_{+,M}^* P_M(1) \times_{\rho_{d+1}} \mathbb{R}^{d+1} \xleftarrow{\psi_M(1)} & \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus \mathbb{R} \oplus \mathrm{pr}_{+,M}^* TM \xrightarrow{c_M} & \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus \mathbb{R} \oplus \mathrm{pr}_{+,M}^* TM \\ + & \sigma & \mathrm{id}_{\mathbb{R}\oplus\mathbb{R}}^{d-n}\oplus\sigma^*\mathrm{id}_{TM} & \mathrm{id}_{\mathbb{R}\oplus\mathbb{R}}^{d-n}\oplus(-\mathrm{id}_{\mathbb{R}})\oplus\sigma^*\mathrm{id}_{TM} & \\ + & \mathrm{pr}_{-,M}^* P_M(1) \times_{\rho_{d+1}} \mathbb{R}^{d+1} \xleftarrow{\psi_M(1)} & \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus \mathbb{R} \oplus \mathrm{pr}_{-,M}^* TM & \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus \mathbb{R} \oplus \mathrm{pr}_{-,M}^* TM \\ + & \xi'(1)|_{(-\epsilon/3,0]\times M} & \mathrm{id}_{\mathbb{R}\oplus\mathbb{R}}^{d-n}\oplus\phi' & \mathrm{id}_{\mathbb{R}\oplus\mathbb{R}}^{d-n}\oplus\phi' & \\ + & P_{W'}(1)|_{U'_{\epsilon/3}} \times_{\rho_{d+1}} \mathbb{R}^{d+1} \xrightarrow{\psi_{W'}(1)} & \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus TU'_{\epsilon/3} & (c_{W'}\circ r_{W'})|_{U'_{\epsilon/3}} & \mathbb{R} \oplus \mathbb{R}^{d-n} \oplus TU'_{\epsilon/3} +\end{tikzcd} +$$ + +The upper left square is obviously commutative. The lower left square is +commutative by the $W'$-version of (4.26). The commutativity of the right +square follows from (4.28). Thus the diagram (4.29) is commutative. Under +the identification $T([0, \epsilon/3) \times M) \simeq \mathbb{R} \oplus \text{pr}_{+,M}^* TM$, the composition of the +right vertical arrows in (4.29) corresponds to the isomorphism on the stable +frame bundles induced by the diffeomorphism $\phi'|_{(-\epsilon/3,0] \times M} \circ \sigma: [0, \epsilon/3) \times +M \to U'_{\epsilon/3}$. This implies (4) and completes the proof of Claim 4.27. $\square$ + +Claim 4.27 (3) and (4) allow us to define the geometric stable tangential +$G$-structure on the compact $n$-dimensional $\langle 1\rangle$-manifold $W\cup_M W'$ by gluing +$g_W(1)_{\text{def}}$ and $g_{W'}(1)_{\text{def,rev}}$ along $M$. We denote this geometric stable tangen- +tial $G$-structure by $g_{W\cup W'}$. Note that $f_W\cup f_{W'}\in C^\infty((W\cup_M W', \partial(W\cup_M W'))$. +By the integrality assumption $\text{cw}(\omega)\in\text{Hom}(\Omega_n^G(X,A),\mathbb{Z})$, +we have + +$$ +\int_{W \cup_M W'} c w_{g_W \cup W'} ((f_W \cup f_{W'})^* \omega) \in \mathbb{Z}. +$$ + +By Claim 4.27 (1) and (2), we have + +$$ +\int_{W \cup_M W'} c w_{g_W \cup W'} ((f_W \cup f_{W'})^* \omega) = \int_W c w_{g_W}(f_W^* \omega) - \int_{W'} c w_{g_{W'}}(f_{W'}^* \omega). +$$ + +This implies (4.23) and completes the proof of Lemma 4.22. □ +---PAGE_BREAK--- + +*Proof of Proposition 4.20.* The composition at $\operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R})$ is zero by Lemma 4.17. The other compositions are obviously zero. + +First we show the exactness at $\operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}/\mathbb{Z})$. Suppose that $h \in \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}/\mathbb{Z})$ satisfies $I((0, h)) = 0$. Then there exists $\alpha \in \Omega_{\mathrm{clo}}^{n-1}(X, A; V_{I\Omega^G}^\bullet)/\mathrm{Im}(d)$ with $h = \mathrm{cw}(\alpha)$. Since the homomorphism $\mathrm{cw}(\alpha)$ lifts to an $\mathbb{R}$-valued homomorphism defined by the same formula as (4.14), we see that $h$ is in the image from $\operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R})$. + +Next we show the exactness at $(I\Omega_{\mathrm{dR}}^G)^n(X, A)$. Suppose that $I((\omega, h)) \in (I\Omega_{\mathrm{dR}}^G)^n(X, A)$ satisfies $[\omega] = 0$. There exists $\alpha \in \Omega^{n-1}(X, A; V_{I\Omega^G}^\bullet)$ such that $\omega = d\alpha$. Thus we have $I((\omega, h)) = I((0, h - \mathrm{cw}(\alpha)))$. This implies $p(h - \mathrm{cw}(\alpha)) = I((\omega, h))$. + +Finally we show the exactness at $\operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R})$. It is equivalent to the claim that $\mathrm{ch}: (I\Omega_{\mathrm{dR}}^G)^n(X, A) \to \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{Z})$ is surjective. Take any element in $\operatorname{Hom}(\Omega_n^G(X, A), \mathbb{Z}) \subset \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}) \simeq H^n(X, A; V_{I\Omega^G}^\bullet)$ and take a representative $\omega \in \Omega_{\mathrm{clo}}^n(X, A; V_{I\Omega^G}^\bullet)$. We would like to find a group homomorphism $h: C_{\infty,n-1}^G(X, A) \to \mathbb{R}/\mathbb{Z}$ which satisfies the compatibility condition in Definition 4.11 (1) (c) with $\omega$. + +The compatibility condition with $\omega$ already determines the value of $h$ on the kernel of the forgetful map $C_{\infty,n-1}^G(X, A) \to \Omega_{n-1}^G(X, A)$. Namely, given a geometric smooth stable tangential $G$-cycle $(M, g_M, f_M)$ over $(X, A)$ of dimension $(n-1)$ which is null-bordant, take any bordism data $(W, g_W, f_W)$ for $(M, g_M, f_M)$. Then set + +$$h([M, g_M, f_M]) := \int_W \mathrm{cw}_{g_W}(f_W^*\omega) \quad (\text{mod } \mathbb{Z}).$$ + +Well-definedness follows from Lemma 4.22. This induces a group homomorphism $h$ on the kernel of the forgetful map $C_{\infty,n-1}^G(X, A) \to \Omega_{n-1}^G(X, A)$. Since $\mathbb{R}/\mathbb{Z}$ is an injective group, there exists a group homomorphism $h: C_{\infty,n-1}^G(X, A) \to \mathbb{R}/\mathbb{Z}$ extending it, so we get the result. $\square$ + +In Subsection 4.1.3 we extend the functor $(I\Omega_{\mathrm{dR}})^*$ to topological spaces, and in Subsection 4.3 we show that it is a generalized cohomology theory. The following result, combined with those results, implies that the quadruple $((I\Omega_{\mathrm{dR}}^G)^*, R, I, a)$ is its differential extension in the sense of Definition 2.57 (Proposition 4.122). + +**Proposition 4.30.** Let $(X, A)$ be a pair of manifolds. + +(1) We have $R \circ a = d$. + +(2) For any pair of manifolds $(X, A)$, the following diagram commutes. + +(3) For any pair of manifolds $(X, A)$, the following sequence is exact. + +(4.31) + +$$((I\Omega_{\mathrm{dR}}^G)^{-1}(X, A)) \xrightarrow{\mathrm{ch}} \Omega^{*-1}(X, A; V_{I\Omega^G}^\bullet)/\mathrm{Im}(d) \xrightarrow{a} ((I\widehat{\Omega}_{\mathrm{dR}}^G)^*(X, A)) \xrightarrow{I} ((I\Omega_{\mathrm{dR}}^G)^*(X, A) \to 0.$$ +---PAGE_BREAK--- + +*Proof.* (1) and (2) are obvious. For (3), the exactness at $\Omega^{*-1}(X, A; V_{\widehat{I\Omega^G}}^\bullet)/\text{Im}(d)$ easily follows from the exactness of (4.21) at $\text{Hom}(\Omega_n^G(X, A), \mathbb{R})$. The remaining parts are exact by definition. $\square$ + +Recall that, in general, for a generalized cohomology theory we have the $S^1$-integration maps (2.9). Our differential model $(\widetilde{\text{I}\Omega}_{\text{dR}}^G)^*$ has a refinement of it, as follows. As shown in Proposition 4.30, it makes our model a *dif-ferential extension with $S^1$-integration* in the sense of [BS10, Definition 1.3] (see Definition 2.62). + +First we define the following geometric stable tangential structure on $S^1 \times M$. Let $M$ be an $n$-dimensional $\langle k \rangle$-manifold and $g = (d, P, \nabla, \psi)$ be a geometric stable tangential $G$-structure with $d \ge n + 2$ on $M$. For the 2-dimensional disk $D^2 = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 \le 1\}$, let $g_{D^2} := (d, \text{pr}_M^*P, \text{pr}_M^*\nabla, \text{pr}_M^*\psi)$ be the geometric stable tangential structure on $D^2 \times M$ defined by using the identification $\mathbb{R}^{d-n-2} \oplus T(D \times M) \simeq \text{pr}_M^*(\mathbb{R}^{d-n-2} \oplus \mathbb{R}^2 \oplus TM) = \text{pr}_M^*(\mathbb{R}^{d-n} \oplus TM)$. We can take the obvious collar structure near the boundary $\partial D^2 \times M$ which is induced by the polar coordinates $(x, y) = (r \cos \theta, r \sin \theta)$. Then we get + +$$ (4.32) \qquad \psi_{S^1} : \text{pr}_M^* P \times_{\rho_d} \mathbb{R}^d \simeq \mathbb{R}^{d-n-1} \oplus T(S^1 \times M) $$ + +such that $(D^2 \times M, g_{D^2}, f \circ \text{pr}_M)$ is a bordism data for $(S^1 \times M, g_{S^1}, f \circ \text{pr}_M)$ for any $f: M \to X$, where $g_{S^1} := (d, \text{pr}_M^*P, \text{pr}_M^*\nabla, \psi_{S^1})$. + +**Definition 4.33** (The bounding geometric stable tangential structure). Let $M$ be an $n$-dimensional $\langle k \rangle$-manifold and $g = (d, P, \nabla, \psi)$ be a geometric stable tangential $G$-structure with $d \ge n+2$ on $M$. The bounding geometric stable tangential structure on $S^1 \times M$ is + +$$ g_{S^1} := (d, \mathrm{pr}_M^* P, \mathrm{pr}_M^* \nabla, \psi_{S^1}), $$ + +where $\psi_{S^1}$ is defined in (4.32). + +*Remark 4.34.* The pullback $i^*\psi_{S^1}$ of $\psi_{S^1}$ from $S^1 \times M$ to $D^1 \times M$ by the map $i: (D^1, S^0) \to (S^1, \{\text{pt}\})$ is homotopic to $\text{pr}_M^*\psi$ on $D^1 \times M$ which consists the structure $g_{D^1}$ in (3.3). + +**Definition 4.35** (*The $S^1$-integration map for $(\widetilde{\text{I}\Omega}_{\text{dR}}^G)^*$*. ) Let $n \in \mathbb{Z}_{\ge 0}$, we define the following map natural in $(X, A)$, + +$$ \int : (\widehat{\text{I}\Omega}_{\text{dR}}^G)^{n+1} (S^1 \times (X, A)) \to (\widehat{\text{I}\Omega}_{\text{dR}}^G)^n (X, A) $$ + +by mapping $(\omega, h)$ to $(\int \omega, \int h)$, where (see Remark 4.37) + +• $\int \omega$ is the image of $\omega$ under the $S^1$-integration of differential forms. + +• We define the homomorphism $\int h: C_{\infty,n-1}^G(X,A) \to \mathbb{R}/\mathbb{Z}$ by + +$$ (\int h)([M,g,f]) := -h([S^1 \times M, g_{S^1}, \text{id}_{S^1} \times f]). $$ + +Here $g_{S^1}$ is given by Definition 4.33. +---PAGE_BREAK--- + +The natural transformation $\int$ induces a natural transformation on the topological level, also denoted by + +$$ (4.36) \qquad \int : (I\Omega_{\mathrm{dR}}^G)^{n+1}(S^1 \times -) \to (I\Omega_{\mathrm{dR}}^G)^n(-). $$ + +We call them the $S^1$-integration map for $(\widehat{I\Omega}_{\mathrm{dR}}^G)^*$ and $(I\Omega_{\mathrm{dR}}^G)^*$, respectively. + +*Remark 4.37.* Here we explain that the pair $(\int \omega, \int h)$ in Definition 4.35 satisfies the compatibility condition in Definition 4.11 (1) (c). Assume that we are given a bordism data $(W, g_W, f_W)$ of a geometric smooth stable tangential $G$-cycle $(M, g_M, f_M)$ of dimension $(n-1)$ over $(X, A)$. Then, it is important to note that $(S^1 \times W, (g_W)_{S^1}, \mathrm{id}_{S^1} \times f_W)$ is not a bordism data for $(S^1 \times M, (g_M)_{S^1}, \mathrm{id}_{S^1} \times f_M)$. Instead, writing $g_M = (d, P, \nabla, \psi)$ with $d \ge n+2$ so that $g_{S^1} = (d, \mathrm{pr}_M^*P, \mathrm{pr}_M^*\nabla, \psi_{S^1})$, consider an additional automorphism on the stable tangent bundle, + +$$ r = \mathrm{id}_{\mathbb{R}^{d-n-1}} \oplus \eta \oplus \mathrm{id}_{TM} : \mathbb{R}^{d-n-1} \oplus \mathbb{R}^2 \oplus TM \to \mathbb{R}^{d-n+1} \oplus \mathbb{R}^2 \oplus TM $$ + +where + +$$ (4.38) \qquad \eta := \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}. $$ + +We define another geometric stable tangential $G$-structure $((g_M)_{S^1})' := (d, \mathrm{pr}_M^*P, \mathrm{pr}_M^*\nabla, r \circ \psi_{S^1})$. Then $(S^1 \times W, (g_W)_{S^1}, \mathrm{id}_{S^1} \times f_W)$ is a bordism data for $(S^1 \times M, ((g_M)_{S^1})'$, $\mathrm{id}_{S^1} \times f_M)$. Now the compatibility condition of $(\omega, h)$ on this bordism data tells us that + +$$ (4.39) \qquad \int_{S^1 \times W} \mathrm{cw}_{(g_W)_{S^1}}(\omega) = h([S^1 \times M, ((g_M)_{S^1})', \mathrm{id}_{S^1} \times f_M]). $$ + +By the sign convention [2.61] we have + +$$ (4.40) \qquad \int_{S^1 \times W} \mathrm{cw}_{(g_W)_{S^1}}(\omega) = \int_W \mathrm{cw}_{g_W} \left( \int \omega \right). $$ + +Now notice that the geometric stable tangential $G$-structure $((g_M)_{S^1})'$ on $S^1 \times M$ is related to $((g_M)_{S^1})_{\mathrm{op}}$ as in the assumption in Lemma 4.16. Thus, by that Lemma we have + +$$ (4.41) $$ + +$$ (4.42) \qquad \begin{aligned} h([S^1 \times M, ((g_M)_{S^1})', \mathrm{id}_{S^1} \times f_M]) &= -h([S^1 \times M, (g_M)_{S^1}, \mathrm{id}_{S^1} \times f_M]) \\ &= (\int h)([M, g_M, f_M]). \end{aligned} $$ + +Combining (4.39), (4.40) and (4.41), we get the desired compatibility condition for $(\int \omega, \int h)$. + +### 4.1.2. Remarks on physical applications. +In the context of unitary QFT's in physics, we are usually interested in $G$ with the following properties. First we require that the image of $\rho_d$ contains at least $\mathrm{SO}(d, \mathbb{R})$, + +$$ (4.43) \qquad \mathrm{SO}(d, \mathbb{R}) \subset \rho_d(G_d). $$ +---PAGE_BREAK--- + +Next we require that the following commutative diagram is a pullback dia- +gram, + +$$ +(4.44) \qquad +\begin{tikzcd} +G_d \arrow[r, "p_d"] \arrow[d, "s_d"] & O(d, \mathbb{R}) \arrow[dl] \\ +G_{d+1} \arrow[r, "p_{d+1}"'] & O(d+1, \mathbb{R}) +\end{tikzcd} +. +$$ + +For $G$ satisfying these properties, we define + +**Definition 4.45.** Let $M$ be a $\langle k \rangle$-manifold with a collar structure. A physical tangential $G$-structure on $M$ is a geometric stable tangential structure $g = (d, P, \nabla, \psi)$ with the following additional requirements. + +• $d = \dim M$, so there is no stabilization $\mathbb{R}^{d-\dim M}$ attached to $TM$. + +• We have a Riemannian metric on $TM$ induced from the standard metric on $P \times_{\rho_d} \mathbb{R}^{\dim M}$ by the isomorphism $\psi: P \times_{\rho_d} \mathbb{R}^{\dim M} \simeq TM$. The connection induced on $P \times_{\rho_d} \mathbb{R}^{\dim M} \simeq TM$ from $\nabla$ coincides with the Levi-Civita connection of the Riemannian metric. + +For a pair of manifolds $(X, A)$, we define $C_{ph,n}^G(X, A)$ to be a subgroup of $C_{\infty,n}^G(X, A)$ which is generated by $[M, g, f]$ in which $g$ is a physical tangential $G$-structure on $M$. + +We remark that the 1-fold stabilization $g(1)$ (Definition 3.6) of a physical tangential $G$-structure $g$ is not a physical tangential $G$-structure anymore, but we still have $[M,g(1),f] = [M,g,f]$ in $C_{ph,n}^G(X,A) \subset C_{\infty,n}^G(X,A)$. + +**Definition 4.46** (($I\Omega_{\text{ph}}^G)^*$ and $(I\Omega_{\text{ph}}^G)^*$). Let $(X, A)$ be a pair of manifolds and $n \in \mathbb{Z}_{\ge 0}$. + +(1) Define $(I\widehat{\Omega}_{\text{ph}}^G)^n(X, A)$ to be an abelian group consisting of pairs $(\omega, h_{\text{ph}})$, such that + +(a) $\omega$ is a closed $n$-form $\omega \in \Omega_{\text{clo}}^n(X, A; V_{I\widehat{\Omega}^G}^\bullet)$. + +(b) $h_{\text{ph}}$ is a group homomorphism $h_{\text{ph}} : C_{\text{ph},n-1}^G(X, A) \to \mathbb{R}/\mathbb{Z}$. + +(c) $\omega$ and $h_{\text{ph}}$ satisfy the compatibility condition as in Definition 4.11 + +(1) (c), + +$$ +h_{\mathrm{ph}}([M, g_M, f_M]) = \int_W \mathrm{cw}_{g_W}(f_W^*\omega) \pmod{\mathbb{Z}}. +$$ + +for $(W, g_W, f_W)$ in which $g_W$ is a physical tangential $G$-structure, +and $(W, g_W, f_W)$ is a bordism data for $(M, g_M(1), f_M)$. +Abelian group structure is defined in the obvious way. + +(2) We define a homomorphsim of abelian groups, + +$$ +a: \Omega^{n-1}(X, A; V_{I\Omega_G}^\bullet)/\operatorname{Im}(d) \to (I\widehat{\Omega}_{\text{ph}}^G)^n(X, A) +$$ + +$$ +\alpha \mapsto (d\alpha, \operatorname{cw}(\alpha)). +$$ + +We set + +$$ +(I\widehat{\Omega}_{\text{ph}}^{G})^{n}(X, A) := (I\widehat{\Omega}_{\text{ph}}^{G})^{n}(X, A)/\operatorname{Im}(a). +$$ +---PAGE_BREAK--- + +For $n \in \mathbb{Z}_{<0}$ we set $(\widehat{I\Omega_{ph}^G})^n(X, A) = 0$ and $(I\Omega_{ph}^G)^n(X, A) = 0$. + +As in Definition 4.11 we get contravariant functors, + +$$ (\widehat{I\Omega_{ph}^G})^*, (I\Omega_{ph}^G)^*: \text{MfdPair}^{\text{op}} \to \text{Ab}^\mathbb{Z}. $$ + +*Remark 4.47.* Actually, it is not a priori a physical requirement that $\omega \in \Omega^n_{\text{clo}}(X, A; V^{\bullet}_{I\Omega^G})$. We may just require $\omega \in \Omega^n_{\text{clo}}(X, A; V^{\bullet}_{I\Omega^{G_n}})$. (Notice the difference between $G$ and $G_n$.) However, using $V^{\bullet}_{I\Omega^G}$ rather than $V^{\bullet}_{I\Omega^{G_n}}$ excludes some reflection-nonpositive partition functions. In this paper we just use $V^{\bullet}_{I\Omega^G}$ from the beginning rather than formulating reflection positivity. + +**Proposition 4.48.** For $G$ satisfying (4.43) and (4.44), we have the natural isomorphisms of functors $\text{MfdPair}^{\text{op}} \to \text{Ab}^\mathbb{Z}$, + +$$ (4.49) \qquad (\widehat{I\Omega_{dR}^G})^* \simeq (\widehat{I\Omega_{ph}^G})^*, \quad (I\Omega_{dR}^G)^* \simeq (I\Omega_{ph}^G)^*. $$ + +*Proof.* There is an obvious natural transformation $(\widehat{I\Omega_{dR}^G})^* \to (\widehat{I\Omega_{ph}^G})^*$ since $C^G_{ph,n-1}(X,A) \subset C^G_{\infty,n-1}(X,A)$ and the corresponding statement for bordism data. We show that it is actually an isomorphism by constructing the inverse +$(\widehat{I\Omega_{ph}^G})^n(X,A) \to (\widehat{I\Omega_{dR}^G})^n(X,A).$ + +Let $(M, g, f)$ be a geometric smooth stable tangential $G$-cycle with $g = (d, P, \nabla, \psi)$ and $\dim M = n - 1$. First, we take $\psi'$ which is homotopic to $\psi$ such that the two sub-bundles $\mathbb{R}^{d-n+1}$ and $TM$ in $\mathbb{R}^{d-n+1} \oplus TM$ are orthogonal to each other with respect to the Riemannian metric induced by $\psi' : P \times_{\rho_d} \mathbb{R}^d \simeq \mathbb{R}^{d-n+1} \oplus TM$. Then we see that the frame bundle $P \times_{\rho_d} O(d, \mathbb{R})$ is reduced to an $O(n-1, \mathbb{R})$ bundle $\text{Fr}(TM)$ which is the orthonormal frame bundle of $TM$. We have $P \times_{\rho_d} O(d, \mathbb{R}) \simeq \text{Fr}(TM) \times_i O(d, \mathbb{R})$, where $i: O(n-1, \mathbb{R}) \to O(d, \mathbb{R})$ is the inclusion. Under the map, + +$$ P = P \times_{\mathrm{id}} G_d \to P \times_{\rho_d} O(d, \mathbb{R}) \simeq \mathrm{Fr}(TM) \times_i O(d, \mathbb{R}), $$ + +the inverse image of $\mathrm{Fr}(TM) \subset \mathrm{Fr}(TM) \times_i O(d, \mathbb{R})$ is a $G_{n-1}$-bundle due to +the pullback diagram (4.44). We denote this $G_{n-1}$-bundle as $P_{\mathrm{ph}}$. It satisfies + +$$ P_{\mathrm{ph}} \times_{s_{d-1} \circ \cdots \circ s_{n-1}} G_d \simeq P. $$ + +By construction, we have a bundle isomorphism $\psi_{\mathrm{ph}} : P_{\mathrm{ph}} \times_{\rho_{n-1}} \mathbb{R}^{n-1} \simeq TM$ +such that its $(d-n+1)$-fold stabilization $\psi_{\mathrm{ph}}(d-n+1)$ is isomorphic to +$\psi'$ under the above isomorphism $P_{\mathrm{ph}} \times_{s_{d-1}\circ\cdots\circ s_{n-1}} G_d \simeq P$. + +We take a connection $\nabla_{\mathrm{ph}}$ on $P_{\mathrm{ph}}$ satisfying the condition that it induces +the Levi-Civita connection on $P_{\mathrm{ph}} \times_{\rho_{n-1}} \mathbb{R}^{n-1} \simeq TM$. This is possible, because the condition (4.43) implies that the Lie algebra $\mathfrak{g}_{n-1}$ of $G_{n-1}$ contains +the Lie algebra of $SO(n-1, \mathbb{R})$ as a direct summand and hence we can take +any connection on this direct summand. We take it so that it coincides with +the Levi-Civita connection. Then $g_{\mathrm{ph}} := (n-1, P_{\mathrm{ph}}, \nabla_{\mathrm{ph}}, \psi_{\mathrm{ph}})$ is a physical +tangential $G$-structure on $M$. + +By construction, the stabilization $g_{\mathrm{ph}}(d - n + 1)$ is homotopic to the +original smooth stable tangential $G$-structure $g$. Thus we can take a bordism +data $([0, 1] \times M, \tilde{g}, \text{pr}_M^*f)$ for $(M, (g_{\mathrm{ph}}(d - n + 1))_{\text{op}}, f) \sqcup (M, g, f)$. + +Given $(\omega, h_{\mathrm{ph}}) \in (\widehat{I\Omega}_{\mathrm{ph}}^G)^n(X, A)$, we define + +$$ H(\omega, h_{\mathrm{ph}}) : C^G_{\infty, n-1}(X, A) \to \mathbb{R}/\mathbb{Z} $$ +---PAGE_BREAK--- + +by + +$$H(\omega, h_{\text{ph}})([M, g, f]) := h_{\text{ph}}([M, g_{\text{ph}}, f]) + \int_{[0,1]\times M} \operatorname{cw}_{\tilde{g}}((f \circ \text{pr}_M)^*\omega) \pmod{\mathbb{Z}}$$ + +where $g_{\text{ph}}$ and $\tilde{g}$ are constructed as above. The definition of $H(\omega, h_{\text{ph}})$ does not depend on how to take $\tilde{g}$ and $g_{\text{ph}}$ by the following reason. If we have two bordism data ($[0, 1] \times M, \tilde{g}, \text{pr}_M^*f$) and ($[0, 1] \times M, \tilde{g}', \text{pr}_M^*f$), we can glue them as in the proof of Lemma 4.22. Then we further deform the geometric stable tangential $G$-structure on the glued manifold to make it into a physical tangential $G$-structure by the same procedure as above. Let $\tilde{g}_{\text{ph}}$ be the physical tangential $G$-structure on the glued manifold. The evaluation of $\operatorname{cw}_{\tilde{g}_{\text{ph}}}((f \circ \text{pr}_M)^*\omega)$ on the glued manifold is then an integer by the compatibility condition in Definition 4.46. It is also the same as the one before the deformation. Thus we get + +$$\int_{[0,1]\times M} \operatorname{cw}_{\tilde{g}}((f \circ \text{pr}_M)^*\omega) = \int_{[0,1]\times M} \operatorname{cw}_{\tilde{g}'}((f \circ \text{pr}_M)^*\omega) \pmod{\mathbb{Z}}.$$ + +The independence of $H(\omega, h_{\text{ph}})$ on how to take $g_{\text{ph}}$ is also shown by using the compatibility condition for $(\omega, h_{\text{ph}})$. The point is that, given two $g_{\text{ph}}$ and $g'_{\text{ph}}$, a homotopy between $g_{\text{ph}}(d-n+1)$ and $g'_{\text{ph}}(d-n+1)$ can be deformed to a homotopy between $g_{\text{ph}}(1)$ and $g'_{\text{ph}}(1)$ by a physical tangential $G$-structure. + +Clearly the map + +$$ +\begin{gather*} +(\widehat{\Omega}_{\text{ph}}^{G})^{n}(X, A) \to (\widehat{\Omega}_{\text{DR}}^{G})^{n}(X, A) \\ +(\omega, h_{\text{ph}}) \mapsto (\omega, H(\omega, h_{\text{ph}})) +\end{gather*} +$$ + +gives the desired inverse to $(\widehat{\Omega}_{\text{DR}}^G)^n(X, A) \to (\widehat{\Omega}_{\text{ph}}^G)^n(X, A)$. This completes the proof of the natural isomorphism $(\widehat{\Omega}_{\text{DR}}^G)^* \simeq (\widehat{\Omega}_{\text{ph}}^G)^*$. The natural isomorphism $(I\Omega_{\text{DR}}^G)^* \simeq (I\Omega_{\text{ph}}^G)^*$ also follows immediately. $\square$ + +**4.1.3. The singular models.** Next, we define the singular model $(I\Omega_{\text{sing}}^G)^*$. The construction is analogous to that for $(I\Omega_{\text{DR}}^G)^*$, using singular cochains instead of differential forms. The advantage of this model is that it works for any topological space. On the category of manifolds, we have a natural isomorphism between those models as shown in Proposition 4.70. + +First we introduce the smooth singular version of the Chern-Weil construction in Definition 4.4. + +**Definition 4.50.** (1) Let $W$ be a manifold and $(P, \nabla)$ be a principal $G_d$-bundle with connection on it. Recall that we have a homomorphism $\operatorname{cw}\nabla: V_{I\Omega^G}^n \to \Omega_{clo}^n(W; P \times_{G_d} \mathbb{R}_{G_d})$ (4.6). Regarding the differential forms as smooth singular cochains and extending it $C^\infty(W; \mathbb{R})$-linearly, we get a homomorphism + +$$ (4.51) \qquad \operatorname{cw}\nabla : C_\infty^*(W; V_{I\Omega^G}^\bullet) \to C_\infty^*(W; P \times_{G_d} \mathbb{R}_{G_d}), $$ + +which preserves the total degree and restricts to a homomorphism $\operatorname{cw}\nabla: Z_\infty^*(W; V_{I\Omega^G}^\bullet) \to Z_\infty^*(W; P \times_{G_d} \mathbb{R}_{G_d})$. Explicitly, given $\lambda \in$ +---PAGE_BREAK--- + +$C_{\infty}^{*}(W; V_{I^{\Omega G}}^{\bullet})$, we apply (4.5) on the coefficient and denote the resulting element by $\lambda_d$. Write as a finite sum $\lambda_d = \sum_i \xi_i \otimes \phi_i$ with $\xi_i \in C_{\infty}^{*}(X, A; \mathbb{R})$ and $\phi_i \in V_{I^{\Omega G}_d}^{\bullet}$. This is possible because $V_{I^{\Omega G}_d}^{\bullet}$ is finite dimensional. Then set + +$$ (4.52) \qquad \mathrm{cw}\nabla(\lambda) := \sum_i \xi_i \cup \mathrm{cw}\nabla(\phi_i) $$ + +(2) If $W$ be a manifold with a geometric stable tangential $G$-structure $g = (d, P, \nabla, \psi)$, the isomorphism $\psi$ induces the isomorphism $\psi: P \times_{G_d} \mathbb{R}_{G_d} \simeq \text{Ori}(W)$. Composing the homomorphism (4.51) with this $\psi$ on the coefficient, we define a $C_{\infty}^{*}(W; \mathbb{R})$-linear homomorphism, + +$$ (4.53) \qquad \mathrm{cw}_g := \psi \circ \mathrm{cw}\nabla : C_{\infty}^{*}(W; V_{I^{\Omega G}}^{\bullet}) \to C_{\infty}^{*}(W; \mathrm{Ori}(W)), $$ + +which preserves the total degree and restricts to the homomorphism +$\mathrm{cw}_g : Z_{\infty}^{*}(W; V_{I^{\Omega G}}^{\bullet}) \to Z_{\infty}^{*}(W; \mathrm{Ori}(W))$. + +**Remark 4.54.** A similar remark to Remark 4.10 also applies here. The homomorphism $\mathrm{cw}_g$ in Definition 4.50 (2) actually depends on $\psi$ in $g = (d, P, \nabla, \psi)$ only through its *homotopy class*. + +Suppose we are given a pair of topological spaces $(X, A)$ and an element $\lambda \in C^*(X, A; V_{I^{\Omega G}}^{\bullet})$. If $W$ is a manifold equipped with a geometric stable tangential $G$-structure $g$, given a map $f \in \text{Map}((W, \emptyset), (X, A))$, regarding the pullback $f^*\lambda \in C^*(W, f^{-1}(A); V_{I^{\Omega G}}^{\bullet})$ as a smooth singular cochain, the homomorphism (4.9) gives + +$$ \mathrm{cw}_g(f^*\lambda) \in C_\infty^*(W, f^{-1}(A); \mathrm{Ori}(W)). $$ + +We remark that, even though $\lambda$ is a singular cochain, the resulting $\mathrm{cw}_g(f^*\lambda)$ is only a *smooth* singular cochain. If $\lambda \in Z^n(X, A; V_{I^{\Omega G}}^{\bullet})$, the above construction induces a homomorphism + +$$ \mathrm{cw}(\lambda): \Omega_n^G(X, A) \to \mathbb{R}, \quad [(W, g, t, f)] \mapsto \langle \mathrm{cw}_g(f^*\lambda), t \rangle_W. $$ + +This gives the isomorphism $H^n(X, A; V_{I^{\Omega G}}^{\bullet}) \simeq \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R})$. + +When $(X, A)$ is a pair of manifolds, $\lambda \in C_{\infty}^{*}(X, A; V_{I^{\Omega G}}^{\bullet})$ and $f \in C^{\infty}((W, \emptyset), (X, A))$, by the same construction we get $\mathrm{cw}_g(f^*\lambda) \in C_{\infty}^{*}(W, f^{-1}(A); \mathrm{Ori}(W))$. + +**Remark 4.55.** The Chern-Weil constructions in the differential model (Definition 4.4) and the (smooth) singular model (Definition 4.50) are not compatible with the map regarding differential forms as smooth singular cochains, i.e., if $\omega \in \Omega^*(W; V_{I^{\Omega G}}^{\bullet})$, the results of applying (4.7) and (4.51) are different. This is because we used the wedge product in (4.8) and the cup product in (4.52). We use the same notation when there is no fear of confusion, with the understanding that we use Definition 4.4 when we are working in differential models and Definition 4.50 when we are working in (smooth) singular models. We will deal with this difference in the proof of Proposition 4.70. + +**Definition 4.56** ($(\widetilde{I^{\Omega_G}})^*$ and $(I_{\text{sing}}^G)^*$). Let $(X, A)$ be a pair of topological spaces. Let $n \in \mathbb{Z}_{\ge 0}$. + +(1) Define $(\widetilde{I_{\text{sing}}^G})^n(X, A)$ to be an abelian group consisting of pairs $(\lambda, h)$, such that + +(a) $\lambda$ is a singular $n$-cocycle $\lambda \in Z^n(X, A; V_{I^{\Omega G}}^{\bullet})$. +---PAGE_BREAK--- + +(b) $h$ is a group homomorphism $h: C^G_{\Delta, n-1}(X, A) \to \mathbb{R}/\mathbb{Z}$. + +(c) $\lambda$ and $h$ satisfy the following compatibility condition. Assume that we are given a bordism data $(W, g_W, t_W, f_W)$ of a geometric singular stable tangential $G$-cycle $(M, g_M, t_M, f_M)$ of dimension $(n-1)$ over $(X, A)$ (Definition 3.13). Then we have + +$$h([M, g_M, t_M, f_M]) = \langle \mathrm{cw}_g(W)(f_W^\lambda), t_W \rangle_W \pmod{\mathbb{Z}}.$$ + +Abelian group structure is defined in the obvious way. + +(2) We define a homomorphsim of abelian groups, + +$$ (4.57) \qquad a_{\text{sing}} : C^{n-1}(X, A; V_{I^{\Omega_G}}^{\bullet}) \to (\widehat{I^{\Omega_{\text{dR}}^G}})^n(X, A) $$ + +$$ (4.58) \qquad \alpha \mapsto (\delta\alpha, \operatorname{cw}(\alpha)). $$ + +Here the homomorphism $\operatorname{cw}(\alpha): C^G_{\Delta, n-1}(X, A) \to \mathbb{R}/\mathbb{Z}$ is defined by + +$$ (4.59) \qquad \operatorname{cw}(\alpha)([M, g, t, f]) := \langle t, \operatorname{cw}_g(f^*\alpha) \rangle_M \pmod{\mathbb{Z}}. $$ + +We set + +$$ (\widehat{I\Omega}_{\text{sing}}^G)^n(X, A) := (\widehat{I\Omega}_{\text{sing}}^G)^n(X, A)/\text{Im}(a_{\text{sing}}). $$ + +For $n \in \mathbb{Z}_{<0}$ we set $(\widehat{I\Omega}_{\text{sing}}^G)^n(X, A) = 0$ and $(\widehat{I\Omega}_{\text{sing}}^G)^n(X, A) = 0$. + +For a map $\phi \in \operatorname{Map}((X, A), (Y, B))$ between two pairs of spaces, by the pullback we get the homomorphisms from $\phi^*:(\widehat{I\Omega}_{\text{sing}}^G)^n(Y, B) \to (\widehat{I\Omega}_{\text{sing}}^G)^n(X, A)$ and $\phi^*: (\widehat{I\Omega}_{\text{sing}}^G)^n(Y, B) \to (\widehat{I\Omega}_{\text{sing}}^G)^n(X, A)$. Thus we get contravariant func-tors, + +$$ (\widehat{I\Omega}_{\text{sing}}^G)^*, (\widehat{I\Omega}_{\text{sing}}^G)^* : \text{TopPair}^{\text{op}} \to \text{Ab}^{\mathbb{Z}}. $$ + +When $(X, A)$ is a pair of manifolds, we can also consider the “smooth singular versions” $(\widehat{I\Omega}_{\text{sing},\infty}^G)^*(X, A)$ and $(\widehat{I\Omega}_{\text{sing},\infty}^G)^*(X, A)$ of the groups defined in Definition 4.56. Namely, in the definition we use smooth singu-lar cochains and cocycles instead of singular ones, $C^G_{\Delta,\infty,n-1}(X,A)$ instead of $C^G_{\Delta,n-1}(X,A)$, and require $f_W$ to be smooth. This gives contravariant functors, + +$$ (\widehat{I\Omega}_{\text{sing},\infty}^G)^*, (\widehat{I\Omega}_{\text{sing},\infty}^G)^* : \text{MfdPair}^{\text{op}} \to \text{Ab}^{\mathbb{Z}}. $$ + +As in the case of differential models, we use the notation $(\widehat{IZ}_{\text{sing}})^* := (\widehat{I\Omega}_{\text{sing}}^{\mathrm{fr}})^*, (\widehat{IZ}_{\text{sing}})^* := (\widehat{I\Omega}_{\text{sing},\infty}^{\mathrm{fr}})^*, (\widehat{IZ}_{\text{sing},\infty}^*) := (\widehat{I\Omega}_{\text{sing},\infty}^{\mathrm{fr}})^*$ and $(\widehat{IZ}_{\text{sing},\infty})^* := (\widehat{I\Omega}_{\text{sing},\infty}^{\mathrm{fr}})^*$. + +**Definition 4.60** (Structure maps for $\widehat{I\Omega}_{\text{sing}}^G$ and $\widehat{I\Omega}_{\text{sing}}^G$). We define the following maps natural in $(X, A)$. The well-definedness is easy by Definition 4.56. + +• We denote the quotient map by + +$$ I_{\text{sing}} : (\widehat{\Omega}_{\text{sing}}^G)^*(X, A) \to (\widehat{\Omega}_{\text{sing}}^G)^*(X, A). $$ + +• We define + +$$ \operatorname{ch}_{\text{sing}} : (\widehat{I\Omega}_{\text{sing}}^G)^*(X, A) \to H^n(X, A; V_{I^{\Omega_G}}^\bullet) (\simeq \operatorname{Hom}(\Omega_*^G(X, A), \mathbb{R})), \quad I_{\text{sing}}((\lambda, h)) \mapsto [\lambda](\mapsto \operatorname{cw}(\lambda)). $$ +---PAGE_BREAK--- + +• We define + +$$p_{\text{sing}} : \operatorname{Hom}(\Omega_{*-1}^G(X, A), \mathbb{R}/\mathbb{Z}) \to (I\Omega_{\mathrm{dR}}^G)^*(X, A), \quad h \mapsto I_{\text{sing}}((0, h)),$$ + +where we regard $h \in \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}/\mathbb{Z})$ as a group homomorphism $h: C_{\Delta,n-1}^G(X,A) \to \mathbb{R}/\mathbb{Z}$. + +The corresponding structure maps for smooth singular versions are defined analogously and denoted by $I_{\text{sing},\infty}$, $\text{ch}_{\text{sing},\infty}$ and $p_{\text{sing},\infty}$. + +The analogue of Lemma 4.16 is the following. + +**Lemma 4.61.** Let $(X, A)$ be a pair of topological spaces, $n$ be a positive integer, and assume we are given an element $(\lambda, h) \in (\widehat{I\Omega}_{\text{sing}}^G)^n(X, A)$. Let $(M, g, t, f)$ be a geometric singular stable tangential $G$-cycle of $(n-1)$-dimension over $(X, A)$ with $g = (d, P, \nabla, \psi)$. Assume we change the isomorphism $\psi: P \times_{\rho_d} \mathbb{R}^d \simeq \mathbb{R}^{d-n} \oplus TM$ to a homotopic one, $\psi'$, and denote the resulting geometric stable tangential $G$-structure on $M$ by $g' = (d, P, \nabla, \psi')$. Then we have + +$$h([M, g, t, f]) = h([M, g', t, f]).$$ + +*Proof.* The proof is the same as that for Lemma 4.16, where now we use a bordism data of the form $([0, 1] \times M, \tilde{g}, P_{[0,1]}t, f \circ \text{pr}_M)$. $\square$ + +**Definition 4.62** (The coboundary map for $I\Omega_{\text{sing}}^G$). For a pair of topological spaces $(X, A)$ and $n \in \mathbb{Z}_{\ge 0}$, we define a map + +$$\widehat{\delta^n} : (\widehat{I\Omega}_{\text{sing}}^G)^n(A) \to (\widehat{I\Omega}_{\text{sing}}^G)^{n+1}(X, A)$$ + +by mapping $(\lambda, h)$ to $(\delta\bar{\lambda}, -h \circ \partial + \operatorname{cw}(\bar{\lambda}))$ (see Remark 4.63), where + +• For $\lambda \in Z^n(A; V_{I\Omega^G}^\bullet)$, we denote by $\bar{\lambda} \in C^n(X; V_{I\Omega^G}^\bullet)$ its zero extension, i.e., the singular chain which extends $\lambda$ and takes zero on singular chains which is not in $A$. + +• $h \circ \partial$ is the composition of $\partial: C_{\Delta,n}^G(X,A) \to C_{\Delta,n-1}^G(A)$ in (3.11) with $h$. + +Then, $\widehat{\delta}^n$ induces a homomorphism + +$$\delta^n : (I\Omega_{\text{sing}}^G)^n(A) \to (I\Omega_{\text{sing}}^G)^{n+1}(X, A).$$ + +Indeed, for $\alpha \in C^{n-1}(A; V_{I\Omega^G}^\bullet)$, we have $\bar{\delta}\alpha - \delta\bar{\alpha} \in C^n(X, A; V_{I\Omega^G}^\bullet)$ and + +$$ +\begin{align*} +\widehat{\delta}^n(a_{\text{sing}}(\alpha)) &= (\delta\bar{\delta}\alpha, -\operatorname{cw}(\alpha) \circ \partial + \operatorname{cw}(\bar{\delta}\alpha)) \\ +&= (\delta(\bar{\delta}\alpha - \delta\bar{\alpha}), \operatorname{cw}(\bar{\delta}\alpha - \delta\bar{\alpha})) \\ +&= a_{\text{sing}}(\bar{\delta}\alpha - \delta\bar{\alpha}). +\end{align*} +$$ + +We call $\delta$ the *coboundary map* for $I\Omega_{\text{sing}}^G$. + +*Remark 4.63.* In the above definition of $\widehat{\delta}^n$, the fact that the pair $(\delta\bar{\lambda}, -h \circ \partial + \operatorname{cw}(\bar{\lambda}))$ satisfies the compatibility condition in $(\widehat{I\Omega}_{\text{sing}}^G)^{n+1}(X, A)$ can be checked as follows. Take a geometric singular stable tangential $G$-cycle $(M, g_M, t_M, f_M)$ of dimension $n$ over $(X, A)$ and its bordism data $(W, g_W, t_W, f_M)$ with $\varphi: \partial_0 W \simeq M$. We have a diffeomorphism $\partial\partial_0 W \simeq \partial\partial_1 W$ as manifolds. However, the geometric singular stable tangential $G$-structure on $\partial M$ +---PAGE_BREAK--- + +which is induced as a boundary of ∂₁W differs from the one induced as a +boundary of ∂₀W by an additional automorphism + +$$ +r = \mathrm{id}_{\mathbb{R}^{d-n-1}} \oplus \eta \oplus \mathrm{id}_{T(\partial M)} : \mathbb{R}^{d-n-1} \oplus \mathbb{R}^2 \oplus T(\partial M) \to \mathbb{R}^{d-n+1} \oplus \mathbb{R}^2 \oplus T(\partial M) +$$ + +where η is the flip as in (4.38). In particular, it reverses the orientation of $\mathbb{R}^{d-n-1} \oplus \mathbb{R}^2 \oplus T(\partial M)$, and the resulting geometric singular stable tangential $G$-structure is related to $(g_M)_{\text{op}}$ as in the assumption of Lemma 4.61. Thus we get + +$$ +h \circ \partial([M, g_M, t_M, f_M]) = -\langle \mathrm{cw}_{g_W}(f_W^* \lambda), t_{\partial_1 W} \rangle, +$$ + +where we have used the compatibility condition of $(\lambda, h) \in (\widehat{\Omega}_{\text{sing}}^G)^n(A)$ on $\partial_1 W$ whose fundamental chain is taken as $t_{\partial_1 W} := \partial t_W - \varphi^* t_M$, and the minus sign is from $h([M, (g_M)_{\text{op}}, t_M, f_M]) = -h([M, g_M, t_M, f_M])$ and Lemma 4.61. Using it, we obtain the compatibility condition of $\delta^n(\lambda, h)$. + +In Subsection 4.3, we show that the pair ${}(I\Omega_{\text{sing}}^G)^*, \delta^*}$ defined above is a generalized cohomology theory (Proposition 4.94). The following map is shown to coincide with the $S^1$-integration map given by (2.9) for $I\Omega_{\text{sing}}^G$ in Proposition 4.114. + +**Definition 4.64** (The $S^1$-integration map for $I\Omega_{\text{sing}}^G$). For a pair of topological spaces $(X, A)$ and $n \in \mathbb{Z}_{\ge 0}$, we define a homomorphism + +$$ +\int : (\widehat{I\Omega_{\text{sing}}^G})^{n+1}(S^1 \times (X, A)) \to (\widehat{I\Omega_{\text{sing}}^G})^n(X, A) +$$ + +by mapping $(\lambda, h)$ to $(\int \lambda, \int h)$, where + +* The element $\int \lambda \in Z^n(X, A; V_{I\Omega_G}^\bullet)$ is the image of $\lambda$ under the map $\int : C^{n+1}(S^1 \times (X, A); V_{I\Omega_G}^\bullet) \to C^n(X, A; V_{I\Omega_G}^\bullet)$ defined in (2.53). + +* We define the homomorphism $\int h : C_{\Delta, n-1}^G(X, A) \to \mathbb{R}/\mathbb{Z}$ by + +$$ +\left( \int h \right) ([M, g, t, f]) := -h([S^1 \times M, g_{S^1}, P_{S^1} t, \mathrm{id}_{S^1} \times f]). +$$ + +Here $g_{S^1}$ is given by Definition 4.33. To regard $P_{S^1}t$ (where $P_{S^1}$ is defined in (2.53)) as a fundamental chain on $S^1 \times M$, we use the identification $\text{Ori}(S^1 \times M) \simeq \text{Ori}(M)$ given by $T(S^1 \times M) = TS^1 \oplus TM$ and the sign convention in (2.35) and (2.36). + +The fact that the pair $(\int \lambda, \int h)$ satisfies the compatibility condition is checked in the same way as Remark 4.37. The homomorphism $\int$ induces a homomorphism on the topological level, also denoted by + +$$ +(4.65) \quad \int : (I\Omega_{\text{sing}}^G)^{n+1} (S^1 \times (X, A)) \to (I\Omega_{\text{sing}}^G)^n (X, A). +$$ + +We call it the *S1-integration map* for *IΩsingG*. + +We have the integrality condition for $\lambda$ analogous to Lemma 4.17. The proof is the same. + +**Lemma 4.66** (The integrality condition). *We have* + +$$ +\mathrm{Im}(\mathrm{ch}_{\mathrm{sing}}) \subset \mathrm{Hom}(\Omega_*^G(X, A), \mathbb{Z}), +$$ +---PAGE_BREAK--- + +The group $(I\Omega_{\text{sing}}^G)^n(X, A)$ fits into the same exact sequence as $(I\Omega_{\text{dR}}^G)^n(X, A)$ in Proposition 4.20. + +**Proposition 4.67.** For any pair of topological spaces $(X, A)$ and $n \in \mathbb{Z}_{\ge 0}$, the following sequence is exact. + +$$ (4.68) \qquad \begin{aligned} \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}) &\to \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}/\mathbb{Z}) \xrightarrow{\mathrm{p}_{\text{sing}}} (\mathrm{I}\Omega_{\text{sing}}^G)^n(X, A) \\ &\xrightarrow{\mathrm{ch}_{\text{sing}}} \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}) \to \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}/\mathbb{Z}). \end{aligned} $$ + +When $(X, A)$ is a pair of manifolds, the following sequence is also exact. + +$$ (4.69) \qquad \begin{aligned} \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}) &\to \operatorname{Hom}(\Omega_{n-1}^G(X, A), \mathbb{R}/\mathbb{Z}) \xrightarrow{\mathrm{p}_{\text{sing},\infty}} (\mathrm{I}\Omega_{\text{sing},\infty}^G)^n(X, A) \\ &\xrightarrow{\mathrm{ch}_{\text{sing},\infty}} \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}) \to \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}/\mathbb{Z}). \end{aligned} $$ + +*Proof.* The proof is essentially the same as that of Proposition 4.20. We remark that we use Lemma 2.70 to show the exactness at the fourth term. The details are left to the reader. $\square$ + +Now we show that the differential models in Definition 4.11 are naturally isomorphic to the (smooth) singular models in Definition 4.56, restricted to manifolds. + +**Proposition 4.70.** We have the natural isomorphisms of functors MfdPair$^{OP} \to$ Ab$\mathbb{Z}$, + +$$ (4.71) \qquad (\mathrm{I}\Omega_{\mathrm{dR}}^G)^* \simeq (\mathrm{I}\Omega_{\mathrm{sing},\infty}^G)^* \simeq (\mathrm{I}\Omega_{\mathrm{sing}}^G)^*. $$ + +*Proof.* For the left homomorphism, we construct a natural transformation + +$$ (4.72) \qquad (\widetilde{\mathrm{I}\Omega}_{\mathrm{dR}}^G)^* \to (\widetilde{\mathrm{I}\Omega}_{\mathrm{sing},\infty}^G)^*, $$ + +as follows. Fix a natural cochain homotopy $B$ (2.46) between $\Lambda$ on differential forms and $\cup$ on smooth cochains. The idea is to use the inclusion $\Omega_{\mathrm{clo}}^*(-; V_{I\Omega^G}^\bullet) \hookrightarrow Z_\infty^*(-; V_{I\Omega^G}^\bullet)$ and the forgetful map fgt: $C_{\Delta,\infty,*-1}^G \to C_{\infty,*-1}^G$ in (3.12). But note that, an element $(\omega, h_{\mathrm{dR}}) \in (\widetilde{\mathrm{I}\Omega}_{\mathrm{dR}}^G)^n(X, A)$ does not necessarily satisfy the compatibility condition for the smooth singular model. This is because the Chern-Weil constructions in the differential model (Definition 4.4) and the (smooth) singular model (Definition 4.50) are different (see Remark 4.55). In this proof, we denote the former by cw$^{\mathrm{dR}}$ and the latter by cw$^{\mathrm{sing}}$ to distinguish them. + +To fix this difference, using the natural homotopy $B$ chosen above, we define a homomorphism + +$$ (4.73) \qquad B_\nabla : \Omega^*(W; V_{I\Omega^G}^\bullet) \to C_\infty^{*-1}(W; P \times_{G_d} \mathbb{R}_{G_d}) $$ + +for each manifold $W$ equipped with a principal $G_d$-bundle with connection $(P, \nabla)$ for some $d$, as follows. If the image of $\omega$ in $\Omega^*(W; V_{I\Omega^G d}^\bullet)$ is written as $\omega_d = \sum_i \xi_i \otimes \phi_i$ with $\xi_i \in \Omega^*(W)$ and $\phi_i \in V_{I\Omega^{G_d}}^\bullet$, we set + +$$ (4.74) \qquad B_\nabla(\omega) := \sum_i B(\xi_i, \mathrm{cw}_\nabla(\phi_i)). $$ +---PAGE_BREAK--- + +By the naturality of $B$, the transformation (4.73) is natural in $(W, P, \nabla)$ +and is also compatible with stabilization $d \mapsto (d+1)$. By (2.45), for $\omega \in$ +$\Omega_{\text{clo}}^*(W; V_{I\Omega^G}^\bullet)$ we have + +$$ +(4.75) \qquad \mathrm{cw}_{\nabla}^{\mathrm{dR}}(\omega) - \mathrm{cw}_{\nabla}^{\mathrm{sing}}(\omega) = \delta \mathrm{B}_{\nabla}(\omega). +$$ + +If $W$ be a manifold with a geometric stable tangential $G$-structure $g =$ ($d, P, \nabla, \psi$), applying the associated isomorphism $\psi: P \times_{G_d} \mathbb{R}_{G_d} \simeq \text{Ori}(W)$ on the coefficient of (4.73), we define + +$$ +(4.76) \qquad B_g := \psi \circ B_\nabla : \Omega^*(W; V_{I\Omega^G}^\bullet) \to C_\infty^{*-1}(W; \text{Ori}(W)). +$$ + +Again, if $\omega$ is closed, we have $\mathrm{cw}_g^{\mathrm{dR}}(\omega) - \mathrm{cw}_g^{\mathrm{sing}}(\omega) = \delta B_g(\omega)$. + +Using it, given $(\omega, h_{dR}) \in (\widetilde{I\Omega_{dR}^G})^n(X, A)$, we define + +$$ +(4.77) \qquad h_{\text{sing},\infty} : C_{\Delta,\infty,n-1}^G (X, A) \to \mathbb{R}/\mathbb{Z} +$$ + +by + +$$ +(4.78) \quad h_{\text{sing},\infty}([M, g, t, f]) := h_{\text{dR}}([M, g, f]) - \langle B_g(f^*\omega), t\rangle_M. +$$ + +Then the pair $(\omega, h_{\text{sing},\infty})$ satisfies the compatibility condition for smooth singular model and defines an element in $(I\tilde{\Omega}_{\text{sing},\infty}^G)^n(X, A)$. So we define the natural transformation (4.72) by sending $(\omega, h_{\text{dR}})$ to $(\omega, h_{\text{sing},\infty})$. We easily see that this map induces a natural transformation $(I\tilde{\Omega}_{\text{dR}}^G)^* \to (I\tilde{\Omega}_{\text{sing},\infty}^G)^*$, which does not depend on the choice of $B$ because any two choices of $B$ are naturally cochain homotopic. This homomorphism is compatible with the exact sequences (4.21) and (4.69), so by the five lemma we get the natural isomorphism $(I\tilde{\Omega}_{\text{dR}}^G)^* \simeq (I\tilde{\Omega}_{\text{sing},\infty}^G)^*$. + +For the right, we have the natural transformation, + +$$ +(\widehat{I\Omega_{\text{sing}}^G})^* \rightarrow (\widehat{I\Omega_{\text{sing},\infty}^G})^*, +$$ + +defined using the forgetful map $Z^*(-; V_{I\Omega^G}^\bullet) \to Z_\infty^*(-; V_{I\Omega^G}^\bullet)$ and the inclusion $C_{\Delta,\infty,-1}^G(-) \subset C_{\Delta,*-1}^G(-)$. In this case this naive construction works and induces a natural transformation $(I\Omega_{\text{sing}}^G)^* \to (I\Omega_{\text{sing},\infty}^G)^*$. This homomorphism is also compatible with the exact sequences (4.68) and (4.69), so by the five lemma we get the natural isomorphism $(I\Omega_{\text{sing}}^G)^* \simeq (I\Omega_{\text{sing},\infty}^G)^*$. This completes the proof. $\square$ + +**4.2. Examples of elements in** $\widehat{(I\Omega_{dR}^G)^*}$. In this subsection, we give ex- +amples of elements in $\widehat{(I\Omega_{dR}^G)^*}$ along with the corresponding invertible +QFT's. In this subsection we only list examples. In Section 6 we will give +homotopy-theoretic characterization of some of the examples. + +*Example 4.79 (The holonomy theory (1)).* In this example we consider $G =$ +SO. Fix a manifold $X$ and a hermitian line bundle with unitary connection +$(L, \nabla)$ over $X$. Then we get an element + +$$ +(c_1(\nabla), \operatorname{Hol}\nabla) \in (\widehat{I\Omega_{dR}^{SO}})^2(X). +$$ + +Here, +---PAGE_BREAK--- + +• $c_1(\nabla) = \frac{\sqrt{-1}}{2\pi} F_{\nabla} \in \Omega_{\text{clo}}^2(X)$ is the first Chern form of $\nabla$. Identifying $\mathbb{R}$ with the degree-zero component of $V_{I\Omega^{\text{so}}}^\bullet$, we regard $\Omega_{\text{clo}}^2(X) \subset \Omega_{\text{clo}}^2(X; V_{I\Omega^{\text{so}}}^\bullet)$. + +• The homomorphism $\text{Hol}_{\nabla}: C_{\infty,1}^{\text{SO}}(X) \to \mathbb{R}/\mathbb{Z}$ is given by the holonomy along the closed curve in X. More precisely, an element $[M,g,f]$ in $C_{\infty,1}^{\text{SO}}(X)$ consists of a closed oriented one-dimensional manifold $M$ with a map $f \in C^{\infty}(M,X)$, together with additional information on metric and connection. Regarding it just as an oriented closed curve in $X$, we define $\text{Hol}_{\nabla}([M,g,f])$ to be the holonomy of $(L,\nabla)$ along the curve, by identifying $\mathbb{R}/\mathbb{Z} \simeq U(1)$. + +We will characterize the element $I((c_1(\nabla), \text{Hol}_{\nabla})) \in (I\Omega_{\text{dR}}^{\text{SO}})^2(X)$ in Proposition 6.1. + +*Example 4.80 (The holonomy theory (2))*. In this example we consider $G=\text{SO} \times \text{U}(1)$. Here $\text{U}(1)$ is the *internal symmetry group* explained in Example 2.30 (4). We have an element + +$$ (1 \otimes c_1, \text{Hol}) \in (\widetilde{\Omega}_{\text{dR}}^{\text{SO} \times \text{U}(1)})^2(\text{pt}). $$ + +Here, + +• We have $V_{I\Omega^{\text{SO}\times U(1)}}^{\bullet} = \left( \lim_{d \to 0} (\text{Sym}(\mathfrak{so}(d, \mathbb{R})))^{\text{SO}(d;\mathbb{R})} \otimes_{\mathbb{R}} (\text{Sym}(\mathfrak{u}(1)))^{\text{U}(1)} \right)^{\bullet}$. +The first Chern polynomial $c_1 \in ((\text{Sym}(\mathfrak{u}(1)))^{\bullet})^2$ gives the element $1 \otimes c_1 \in \Omega_{\text{clo}}^2(\text{pt}; V_{I\Omega^{\text{SO}\times U(1)}}^{\bullet}) = V_{I\Omega^{\text{SO}\times U(1)}}^2$. + +• The homomorphism $\text{Hol}: C_{\infty,1}^{\text{SO}\times\mathbf{U}(1)}(\text{pt}) \to \mathbb{R}/\mathbb{Z}$ is given by the holonomy of the internal $\mathbf{U}(1)$-connection. More precisely, an element $[M,g,p_M]$ in $C_{\infty,1}^{\text{SO}\times\mathbf{U}(1)}(\text{pt})$ consists of a closed oriented one-dimensional manifold $M$ with a principal $\mathbf{U}(1)$-bundle with connection, together with other data. We define $\text{Hol}([M,g,p_M])$ to be the holonomy of the $\mathbf{U}(1)$-connection, by identifying $\mathbb{R}/\mathbb{Z} \simeq \mathbf{U}(1)$. + +*Example 4.81 (The classical Chern-Simons theory)*. Fix a compact Lie group $H$ and an element $\lambda \in H^n(BH; \mathbb{Z})$. The corresponding classical Chern-Simons theory ([Fre95], [Fre02]) is an invertible QFT on $(n-1)$-dimensional manifolds equipped with orientations and principal $H$-bundle with connection. This generalizes Example 4.80, which corresponds to $c_1 \in H^2(BU(1); \mathbb{Z})$. Its partition functions are given by the Chern-Simons invariants of $H$-connections. Here we recall its definition. + +Let $\lambda_R \in H^*(BH; \mathbb{R})$ be the $\mathbb{R}$-reduction of the element $\lambda$. Consider the category $\mathcal{C}_H$ of triples $(P, M, \nabla)$, where $P \to M$ is a smooth principal $H$-bundle over a manifold and $\nabla$ is a $H$-connection on $P$. We fix the following data. + +(1) An object $(\mathcal{E}, \mathcal{B}, \nabla_\mathcal{E})$ which is $(n+1)$-classifying, i.e., any object $(P, M, \nabla)$ in $\mathcal{C}_H$ with $\dim M \le n$ admits a morphism to $(\mathcal{E}, \mathcal{B}, \nabla_\mathcal{E})$, and any such morphisms $\phi_1$ and $\phi_2$ are smoothly homotopic. By the theorem of Narasimhan-Ramanan [NR61] such an object exists. + +(2) A differential lift $\tilde{\lambda} \in \hat{H}^n(\mathcal{B}; \mathbb{Z})$ of the element $\lambda \in H^n(\mathcal{B}; \mathbb{Z}) \simeq H^n(BH; \mathbb{Z})$ such that $\tilde{R}(\tilde{\lambda}) = \text{cw}_{\nabla_\mathcal{E}}(\lambda_R) \in \Omega_{\text{clo}}^n(\mathcal{B})$. Here $\hat{H}^n(-; \mathbb{Z})$ +---PAGE_BREAK--- + +is the differential ordinary cohomology group, for example given by +the Cheeger-Simons model explained in Example 2.58. + +In general, for a manifold $X$, an element $\hat{\alpha} \in \hat{H}^n(X; \mathbb{Z})$ associates the higher holonomy functional $\chi_{\hat{\alpha}}$, which assigns an $\mathbb{R}/\mathbb{Z}$-value for a closed $(n-1)$-dimensional manifold $M$ with an orientation $o$ and a smooth map $f: M \to X$. If we use the Cheeger-Simons model $\hat{H}_{CS}^n(X; \mathbb{Z})$ for the differential ordinary cohomology, this is simply given as follows. Write $\hat{\alpha} = (\omega, k)$ in the notation in Example 2.58. Choosing any representative of smooth singular fundamental class $t_M$ of $(M, o)$, we get $f_*t_M \in Z_{\infty,n-1}(X; \mathbb{Z})$. Set + +$$ \chi_{\hat{\alpha}}(M, o, f) := k(f_*t_M) \in \mathbb{R}/\mathbb{Z}. $$ + +This value does not depend on the choice of $t_M$. Now we can define the +Chern-Simons invariants. + +*Definition 4.82 (The Chern-Simons invariants)*. Let $\lambda \in H^n(BH; \mathbb{Z})$ and fix the data (1) and (2) above. Let $M$ be an $(n-1)$-dimensional closed manifold equipped with an orientation $o_M$ and a principal $H$-bundle with connection $(P, \nabla)$. Choose a morphism $\phi: (M, P, \nabla) \to (\mathcal{E}, \mathcal{B}, \nabla_\mathcal{E})$ in $\mathcal{C}_H$. We define the *Chern-Simons invariant* of $(M, o, P, \nabla)$ by + +$$ (4.83) \qquad h_{CS_{\hat{\lambda}}} (M, o, P, \nabla) := \chi_{\hat{\lambda}} (M, o_M, \phi) \in \mathbb{R}/\mathbb{Z}. $$ + +The value (4.83) does not depend on the choice of $\phi$. + +The classical Chern-Simons theory corresponds to the element + +$$ (4.84) \qquad (1 \otimes \lambda_R, h_{CS_{\hat{\lambda}}}) \in (\widehat{I\Omega_{dR}^{SO \times H}})^n(\text{pt}). $$ + +Here $1 \otimes \lambda_R$ is as in Example 4.80, and $h_{CS_{\hat{\lambda}}}$ is regarded as a homomorphism from $C_{\infty,n-1}^{SO \times H}(\text{pt})$. + +Now we analyze the dependence on the choice of a lift $\hat{\lambda}$ of $\lambda$ in (2). By the axioms of differential cohomology (Definition 2.57), we see that two choices $\hat{\lambda}_1$ and $\hat{\lambda}_2$ differs by an element in $H^{n-1}(\mathcal{B}; \mathbb{R}) \simeq H^{n-1}(BH; \mathbb{R})$, i.e., there exists an element $\alpha \in H^{n-1}(\mathcal{B}; \mathbb{R})$ with + +$$ (4.85) \qquad a_{CS}(\alpha) = \hat{\lambda}_1 - \hat{\lambda}_2. $$ + +In particular, if $n$ is even, the lift $\hat{\lambda}$ is unique because $H^{\text{odd}}(BH; \mathbb{R}) = 0$. In general it is possible that the difference (4.85) is nonzero, and in such a case the two elements (4.84) constructed from them are different. But they define the same element in $(I\Omega_{dR}^{SO \times H})^n(\text{pt})$, + +$$ I(1 \otimes \lambda_R, h_{CS_{\hat{\lambda}_1}}) = I(1 \otimes \lambda_R, h_{CS_{\hat{\lambda}_2}}) \in (I\Omega_{dR}^{SO \times H})^n(\text{pt}). $$ + +This is because + +$$ (1 \otimes \lambda_R, h_{CS_{\hat{\lambda}_1}}) - (1 \otimes \lambda_R, h_{CS_{\hat{\lambda}_2}}) = a(1 \otimes \alpha). $$ + +Here the domain of $a$ in (7.4) in this case is $\Omega^{n-1}(\text{pt}; V_{I\Omega^{SO \times H}}^\bullet)/\text{Imd} = (H^*(BSO; \mathbb{R}) \otimes_R H^*(BH; \mathbb{R}))^{n-1}$. Thus we see that, the deformation class + +$$ (4.86) \qquad I(1 \otimes \lambda_R, h_{CS_{\hat{\lambda}}}) \in (I\Omega_{dR}^{SO \times H})^n(\text{pt}). $$ + +is independent of the choice of the lift $\hat{\lambda}$. This element will be characterized +in Proposition 6.3. +---PAGE_BREAK--- + +*Example 4.87 (The theory of massive free complex fermions).* In this example we consider $G = \text{Spin}^c$. Let $k$ be a positive integer. Recall that we have constructed a model $(\widehat{I\Omega_{\text{ph}}^G})^*$ in Definition 4.46 which is isomorphic to $(\widehat{I\Omega_{\text{dR}}^G})^*$ by Proposition 4.48. We are going to construct an element in $(\widehat{I\Omega_{\text{ph}}^{\text{Spin}^c}})^{2k}$ ($-$). + +Given a closed $(2k-1)$-dimensional manifold $M$ with a physical tangential $\text{Spin}^c$-structure $g$ (Definition 4.45), we set + +$$ \bar{\eta}(M, g) := \bar{\eta}(D_M) = \frac{\eta(D_M) + \dim \ker D_M}{2} \in \mathbb{R}. $$ + +where $D_M$ is the Spin$^c$-Dirac operator on $M$ with respect to $g$ and $\eta(D_M) \in \mathbb{R}$ is its eta invariant. Note that we have used the assumption that the connection in $g$ is compatible with the Levi-Civita connection. + +Recall that the Atiyah-Patodi-Singer index theorem ([APS76], [APS75a], [APS75b]) says that, if $(W, g_W)$ is a compact $2k$-dimensional manifold with boundary with a collar structure equipped with a geometric Spin$^c$-structure which is compatible with Levi-Civita connection, we have + +$$ (4.88) \qquad \text{Ind}_{\text{APS}}(D_W) = \int_W \text{Todd}(g_W) - \bar{\eta}(\partial W, \partial g_W). $$ + +Here the left hand side of (4.88) is the Atiyah-Patodi-Singer index of the Dirac operator on $W$, which is an integer. Thus, regarding $\bar{\eta}$ as a homomorphism $\bar{\eta}: C_{\text{ph},2k-1}^{\text{Spin}^c}(\text{pt}) \to \mathbb{R}/\mathbb{Z}$, we get the element + +$$ ((\text{Todd})|_{2k}, \bar{\eta}) \in (\widehat{I\Omega_{\text{ph}}^{\text{Spin}^c}})^{2k}(\text{pt}) \simeq (I\Omega_{\text{ph}}^{\text{Spin}^c})^{2k}(\text{pt}) \simeq (\widehat{I\Omega_{\text{dR}}^{\text{Spin}^c}})^{2k}(\text{pt}). $$ + +This example can be generalized to include target spaces. Fix a manifold $X$ and a hermitian vector bundle with unitary connection $(E, h^E, \nabla^E)$ over $X$. Then, using the eta invariants for Dirac operators twisted by the pullback of $(E, h^E, \nabla^E)$, we get the element + +$$ ((\text{Ch}(\nabla^E) \otimes \text{Todd})|_{2k}, \bar{\eta}_{\nabla^E}) \in (\widehat{I\Omega_{\text{ph}}^{\text{Spin}^c}})^{2k}(X) \simeq (\widehat{I\Omega_{\text{dR}}^{\text{Spin}^c}})^{2k}(X). $$ + +Its deformation class in $(I\Omega_{\text{ph}}^{\text{Spin}^c})^{2k}(X)$ only depends on the class $[E] \in K^0(X)$. + +*Example 4.89 (The theory of massive free real fermions).* Here we consider the real version of Example 4.87. Now $G = \text{Spin}$. We consider the theory on $(8m+3)$-dimension with $m \in \mathbb{Z}_{\ge 0}$, the dimension where the difference from Example 4.87 appears. On Spin manifolds, the Atiyah-Patodi-Singer index theorem (4.88) becomes + +$$ (4.90) \qquad \text{Ind}_{\text{APS}}(D_W) = \int_W \hat{A}(g_W) - \bar{\eta}(\partial W, \partial g_W). $$ +---PAGE_BREAK--- + +Moreover, if $\dim W \equiv 4 \pmod 8$, the APS index is an *even* integer. This +allows us to define the element + +$$ +\left(\frac{1}{2} \hat{A}|_{8m+4}, \frac{1}{2} \bar{\eta}\right) \in \left(\widehat{I \Omega_{\text{ph}}^{\text{Spin}}}\right)^{8m+4} (\text{pt}) \simeq \left(I \Omega_{\text{ph}}^{\text{Spin}}\right)^{8m+4} (\text{pt}) \simeq \left(I \Omega_{\text{dR}}^{\text{Spin}}\right)^{8m+4} (\text{pt}). +$$ + +**4.3. The proof that (IΩsingG)* is a generalized cohomology theory.** + +The goal of this subsection is to show that the pair {⟨$I\Omega_{\text{sing}}^G$⟩*, δ*} in Definition 4.56 and Definition 4.62 is a generalized cohomology theory (Proposition 4.94). First we prove two properties of $h: C_{\Delta,n-1}^G(X,A) \to \mathbb{R}/\mathbb{Z}$ for an element $(\lambda, h) \in (\widehat{I\Omega_{\text{sing}}^G})^n(X, A)$. The first one is the following vanishing property. + +**Lemma 4.91.** For any pair of topological spaces $(X, A)$ and any element $(\lambda, h) \in (\widehat{I\Omega}_{\text{sing}}^{G})^{n}(X, A)$, we have + +$$ +h|_{(A,A)} = 0 : C_{\Delta,n-1}^{G}(A,A) \to \mathbb{R}/\mathbb{Z}. +$$ + +*Proof.* We have $\Omega_{n-1}^G(A, A) = 0$. Thus, for any geometric singular stable tangential $G$-cycle $(M, g_M, t_M, f_M)$ of $(n-1)$-dimension over $(A, A)$, we can find a bordism data $(W, g_W, t_W, f_W)$ over $(A, A)$. Then, the condition in Definition 4.56 (1) (c) applied to $(M, g_M, t_M, f_M)$ and $(W, g_W, t_W, f_W)$ gives + +$$ +h([M, g_M, t_M, f_M]) = \langle cw_{g_W}(f_W^*\lambda), t_W \rangle \pmod{\mathbb{Z}}. +$$ + +But the condition $\lambda \in Z^n(X, A; V_{I\Omega^G}^\bullet)$ implies that $f_W^*\lambda = 0$. Thus $h([M, g_M, t_M, f_M]) = 0$ and the result follows. $\square$ + +Next we show the gluing formula for $h$. + +**Proposition 4.92** (The gluing formula for $h$). Let $(X, A)$ be a pair of topological spaces, $n$ be a positive integer, and assume we are given an element $(\lambda, h) \in (\widetilde{I\Omega}_{\text{sing}}^G)^n(X, A)$. Let $(N, g_N, t_N, f_N)$ be a geometric singular stable tangential $G$-cycle of $(n-2)$-dimension over $(A, \emptyset)$. Assume we have a geometric singular stable tangential $G$-cycle $(M, g, t, f)$ of $(n-1)$-dimension over $(X, A)$ such that its boundary components are divided into $\partial M = \partial^a M \sqcup \partial^b M \sqcup \partial^c M$, and we are given isomorphisms + +$$ +\begin{align*} +\partial^a (M, g, t, f) &\simeq (N, g_N, t_N, f_N) \\ +\partial^b (M, g, t, f) &\simeq (N, (g_N)_{\text{op}}, t_N, f_N). +\end{align*} +$$ + +Then we glue them together along $N$ to get a new geometric singular stable tangential $G$-cycle of $(n-1)$-dimension over $(X, A)$, denoted by + +$(M_{\text{glue}}, g_{\text{glue}}, t_{\text{glue}}, f_{\text{glue}}),$ + + + +with the underlying manifold $M_{\text{glue}} := M/(\partial^a M \sim \partial^b M)$. Then we have + +$$ +(4.93) \qquad h([M,g,t,f]) = h([M_{\text{glue}},g_{\text{glue}},t_{\text{glue}},f_{\text{glue}}]). +$$ + +*Proof.* This is because we can take a bordism data of $(M, g_{\text{op}}, t, f) \sqcup (M_{\text{glue}}, g_{\text{glue}}, t_{\text{glue}}, f_{\text{glue}})$ on which the evaluation of $cw(\lambda)$ vanishes. Such a bordism is essentially given by the cylinder $([0,1] \times M_{\text{glue}}, (g_{\text{glue}})_{[0,1]}, P_{[0,1]}t, f_{\text{glue}} \circ \text{pr}_{M_{\text{glue}}})$, but to be precise we have to make a “hole” on a small neighborhood of $\{0\} \times N$ +---PAGE_BREAK--- + +to make it a $\langle 2 \rangle$-manifold with $\partial_0 = \{0\} \times M \sqcup \{1\} \times M_{\text{glue}}$, and deforming other data accordingly. + +More explicitely, recall that the collar structure at $\partial^a M$ and $\partial^b M$ provides +the identification of $(-\epsilon, \epsilon) \times N$ with a neighborhood of $N$ in $M_{\text{glue}}$ for +some $\epsilon > 0$. We choose a diffeomorphism $\xi': [0, \epsilon) \simeq [\epsilon/2, \epsilon)$. It induces +diffeomorphism $\xi: M \simeq M_{\text{glue}} \setminus ((-\epsilon/2, \epsilon/2) \times N)$, and we put $\xi_*g$ and $\xi_*t$ +on $M_{\text{glue}} \setminus ((-\epsilon/2, \epsilon/2) \times N)$. + +Now let $D := \{x \in [0,1] \times (-\epsilon, \epsilon) \mid \|x\| < \epsilon/2\}$ so that its complement +$[0,1] \times (-\epsilon, \epsilon) \setminus D$ has a structure of $\langle 2 \rangle$-manifold with $\partial_0 = \{0\} \times ((-\epsilon, -\epsilon/2] \cup [\epsilon/2, \epsilon]) \sqcup \{1\} \times (-\epsilon, \epsilon)$. Then $D \times N$ is the desired "hole" in $[0,1] \times M_{\text{glue}}$. +We make $W := ([0,1] \times M_{\text{glue}}) \setminus (V \times N)$ into a desired bordism data as +follows. + +The stable tangential structure on $W$ is given by deforming $(g_{\text{glue}})_{[0,1]}$ on a +small neighborhood of $D \times N$. First recall that by the collar structure of $M$, +the structure $g_{\text{glue}}$ on $(-\epsilon, \epsilon) \times N$ is given by the pullback of a principal $G_d$ +bundle with connection $(P_N, \nabla_N)$ on $N$. Thus we can deform the restriction +of $(g_{\text{glue}})_{[0,1]}$ to $W$ in a way that, the underlying principal bundle $G_d$ with +connection is unchanged (namely given by the pullback of $(P_N, \nabla_N)$ on +$[0,1] \times (-\epsilon, \epsilon) \times N$), and bounds $\xi_*g_{\text{op}}$ and $g_{\text{glue}}$. For the fundamental chain, +choose a smooth homeomorphism $[0, \epsilon/2] \times [0, \epsilon) \to ([0, \epsilon/2] \times [0, \epsilon]) \setminus D$ +which restricts to the above diffeomorphism $\xi': \{0\} \times [0, \epsilon) \simeq \{0\} \times [\epsilon/2, \epsilon)$ and +restricts to the identity on $\{\epsilon/2\} \times [0, \epsilon)$. It induces a map $\tilde{\xi}: [0, \epsilon/2] \times M \to +([0, \epsilon/2] \times M_{\text{glue}}) \setminus (D \times N)$ which restricts to $\xi$ on $\{0\} \times M$. The desired +fundamental chain on $W$ is given by $\tilde{\xi}_* P_{[0,\epsilon/2]} t + P_{[\epsilon/2,1]} t_{\text{glue}}$. Finally we +consider the restriction of $f \circ \text{pr}_M$ on $([0,1] \times M_{\text{glue}}) \setminus (V \times N)$ (recall that +we have required that the map $f$ is compatible with the collar structure +on $M$ (Definition 2.68)). This gives a bordism data of $\xi_*(M, g_{\text{op}}, t, f) \sqcup +(M_{\text{glue}}, g_{\text{glue}}, t_{\text{glue}}, f_{\text{glue}})$. + +By construction the evaluation of cw($\lambda$) on this bordism data is equal to +evaluation of it on $([0,1] \times M_{\text{glue}}, (g_{\text{glue}})_{[0,1]}, P_{[0,1]}t, f_{\text{glue}} \circ \text{pr}_{M_{\text{glue}}})$, which is +zero because it is equal to + +$$ +\begin{flalign*} +& (\operatorname{cw}(g_{\text{glue}})_{[0,1]}, ((f_{\text{glue}} \circ \operatorname{pr}_{M_{\text{glue}}})^*\lambda), P_{[0,1]}t) & = & (\operatorname{cw}_{g_{\text{glue}}}(\tilde{f}_{\text{glue}}^*\lambda), (\operatorname{pr}_{M_{\text{glue}}})_* P_{[0,1]}t) & = & (\operatorname{cw}_{M_{\text{glue}}}), \\ +& & & & & \\ +& \text{and } (\operatorname{pr}_{M_{\text{glue}}})_* P_{[0,1]}t & \text{ is a boundary.} & & & \\ +& & & & & \\ +& & & & & \\ +& & & & & \\ +& & & & & +\end{flalign*} +$$ + +Now we can prove the main result of this subsection. + +**Proposition 4.94.** The pair ${((I\Omega_{\mathrm{sing}}^G)^*, \delta^*)}$ in Definition 4.56 and Definition 4.62 satisfies the Eilenberg-Steenrod axioms. Namely, we have + +(1) The homotopy invariance axiom. If $f: (X, A) \to (Y, B)$ is a homotopy equivalence of CW-pairs, the map + +$$ +f^*: (I\Omega_{\mathrm{sing}}^G)^*(Y, B) \to (I\Omega_{\mathrm{sing}}^G)^*(X, A) +$$ + +is an isomorphism. + +(2) The exactness axiom. For every CW-pair $(X, A)$, the sequence + +$$ +(4.95) \\ +\dots i^* \longrightarrow (I\Omega_{\text{sing}}^G)^{-1}(A) \xrightarrow{\delta^{*-1}} (I\Omega_{\text{sing}}^G)^*(X,A) \xrightarrow{j^*} (I\Omega_{\text{sing}}^G)^*(X) \xrightarrow{i^*} (I\Omega_{\text{sing}}^G)^*(A) \to \dots +$$ + +is exact. +---PAGE_BREAK--- + +(3) The collapse axiom. For every CW-pair $(X, A)$, the collapse $c: (X, A) \to (X/A, \{\text{pt}\})$ induces an isomorphism $c^*: (I\Omega_{\text{sing}}^G)^*(X/A, \{\text{pt}\}) \to (I\Omega_{\text{sing}}^G)^*(X, A)$. + +Moreover it satisfies the additivity axiom. + +(4) The additivity axiom. If $(X, A) = \coprod_i (X_i, A_i)$ is a coproduct in CW-pairs, the induced map + +$$ (I\Omega_{\text{sing}}^{G})^{*}(X, A) \to \prod_{i} (I\Omega_{\text{sing}}^{G})^{*}(X_i, A_i) $$ + +is an isomorphism. + +*Proof.* First note that the functors $(X, A) \mapsto \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R})$ and $(X, A) \mapsto \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{R}/\mathbb{Z})$ are additive generalized cohomology theories, which are represented by the spectra $F(MTG, HR)$ and $F(MTG, IR/2)$, respectively. Thus they satisfy the axioms (1)-(4). Using Proposition 4.67 and the five lemma, we see that $(I\Omega_{\text{sing}}^G)^*$ also satisfies the axioms (1), (3) and (4). + +So we are left to show (2). It is easy to see that the composition of any adjacent two maps are zero. To prove $i^* \circ j^* = 0$, use Lemma 4.91. + +First we show the exactness of (4.95) at $(I\Omega_{\text{sing}}^G)^{n-1}(A)$. Suppose an element $I_{\text{sing}}((\lambda, h)) \in (I\Omega_{\text{sing}}^G)^{n-1}(A)$ satisfies $\delta^{n-1} \circ I_{\text{sing}}((\lambda, h)) = 0$. Then there exists an element $\alpha \in C^{n-1}(X, A; V_{I\Omega^G}^\bullet)$ with $a_{\text{sing}}(\alpha) = \hat{\delta}^{n-1}(\lambda, h)$. This means that we have + +$$ (4.96) \qquad \delta\alpha = \bar{\delta}\lambda, \text{ and} $$ + +$$ (4.97) \qquad \mathrm{cw}(\alpha) = -h \circ \partial + \mathrm{cw}(\bar{\lambda}). $$ + +Note that $\bar{\lambda} - \alpha \in C^{n-1}(X; V_{I\Omega^G}^\bullet)$ restricts to $\lambda$ on $A$ and is closed by (4.96). We construct an element in $(I\Omega_{\text{sing}}^G)^{n-1}(X)$ of the form $(\bar{\lambda} - \alpha, h')$ which lifts $(\lambda, h)$ as follows. First we check the following integrality condition, + +$$ (4.98) \qquad \mathrm{cw}(\bar{\lambda} - \alpha) \in \operatorname{Hom}(\Omega_{n-1}^G(X), \mathbb{Z}). $$ + +Suppose we have an element $[W,g,t,f] \in C_{\Delta,n-1}^G(X)$. We regard it as an element in $C_{\Delta,n-1}^G(X,A)$ with $\partial[W,g,t,f] = \emptyset$, and apply (4.97) to get $\langle \mathrm{cw}_g(f^*(\bar{\lambda}-\alpha)), t \rangle \in \mathbb{Z}$. This implies (4.98). Next we check the following. Assume a geometric singular stable tangential $G$-cycle $(M,g_M,t_M,f_M)$ of dimension $(n-2)$ over $A$ is null-bordant when it is regarded as a geometric singular stable tangential $G$-cycle over $X$. Then for any bordism data $(W,g_W,t_W,g_W)$, we have + +$$ h([M, g_M, t_M, f_M]) = \langle \mathrm{cw}_{g_W}(f_W^*(\bar{\lambda} - \alpha)), t_W \rangle \pmod{\mathbb{Z}}. $$ + +But this follows directly from (4.97), applied to $[W, g_W, t_W, f_W] \in C_{\Delta,n-1}^G(X,A)$. From this and (4.98), we see that the compatibility condition in Definition 4.56 (1) (c) with $\bar{\lambda} - \alpha$ and the condition $h'|_A = h$ already determines the well-defined homomorphism $h'$ on the subgroup of $C_{\Delta,n-2}^G(X)$ generated by $C_{\Delta,n-2}^G(A)$ and the kernel of the forgetful map $C_{\Delta,n-2}^G(X) \to \Omega_{n-2}^G(X)$. Since $\mathbb{R}/\mathbb{Z}$ is an injective group, there exists a group homomorphism $h': C_{\Delta,n-2}^G(X) \to \mathbb{R}/\mathbb{Z}$ extending it. This satisfies the desired properties, and completes the proof of the exactness of (4.95) at $(I\Omega_{\text{sing}}^G)^{n-1}(A). +---PAGE_BREAK--- + +Next we show the exactness of (4.95) at $(I\Omega_{\text{sing}}^G)^n(X, A)$. Suppose an element $I_{\text{sing}}((\lambda, h)) \in (I\Omega_{\text{sing}}^G)^n(X, A)$ satisfies $j^* \circ I_{\text{sing}}((\lambda, h)) = 0$. Then there exists an element $\alpha \in C^{n-1}(X; V_{I\Omega^G}^\bullet)$ with $a_{\text{sing}}(\alpha) = (\lambda, h|_{C_{\Delta,n-1}^G(X)})$ in $(I\Omega_{\text{sing}}^G)^n(X)$. This means that we have + +$$ (4.99) \qquad \delta\alpha = \lambda \text{ in } C^n(X; V_{I\Omega^G}^\bullet), $$ + +$$ (4.100) \qquad h = \operatorname{cw}(\alpha) \text{ on } C_{\Delta, n-1}^G(X). $$ + +Note that we have $\alpha|_A \in Z^{n-1}(A; V_{I\Omega^G}^\bullet)$ by (4.99) and $\alpha - \overline{\alpha}|_A \in C^{n-1}(X, A; V_{I\Omega^G}^\bullet)$. + +We construct an element in $(\widehat{I\Omega_{\text{sing}}^G})^{n-1}(A)$ of the form $(\alpha|_A, h_A)$ such that + +$$ (4.101) \qquad \delta^{n-1}(\alpha|_A, h_A) = (\lambda, h) - a_{\text{sing}}(\alpha - \overline{\alpha}|_A). $$ + +This will imply that $I_{\text{sing}}((\lambda, h))$ is in the image of $\delta^{n-1}$. First we check the following integrality condition, + +$$ (4.102) \qquad \operatorname{cw}(\alpha|_A) \in \operatorname{Hom}(\Omega_{n-1}^G(A), \mathbb{Z}). $$ + +Take any element $[W, g, t, f] \in C_{\Delta, n-1}^G(A)$. By (4.100), we have + +$$ \langle \mathrm{cw}_g(f^*\alpha), t \rangle = h(W, g, t, f) \pmod{\mathbb{Z}}. $$ + +By Lemma 4.91, the right hand side is zero. This implies (4.102). Now, the condition (4.101) is equivalent to the following condition. + +$$ (4.103) \qquad h = -h_A \circ \partial + \operatorname{cw}(\alpha) \text{ on } C_{\Delta, n-1}^G(X, A). $$ + +If an element $[M, g_M, t_M, f_M] \in C_{\Delta, n-2}^G(A)$ is in the image of $\partial: C_{\Delta, n-1}^G(X, A) \to C_{\Delta, n-2}^G(A)$, the condition (4.103) requires us to define $h_A([M, g_M, t_M, f_M])$ by, taking an element $[W, g_W, t_W, f_W] \in C_{\Delta, n-1}^G(X, A)$ with $\partial[W, g_W, t_W, f_W] = [M, g_M, t_M, f_M]$, + +$$ (4.104) \qquad h_A([M, g_M, t_M, f_M]) := -h([W, g_W, t_W, f_W]) + \langle \mathrm{cw}_{g_W}(f_W^*\alpha), t_W \rangle \pmod{\mathbb{Z}}. $$ + +We have to check that (4.104) gives a well-defined homomorphism $h_A$ on the image of $\partial$ which satisfies the compatibility condition in Definition 4.56 (1) (c). For the well-definedness, let $[W', g_{W'}, t_{W'}, f_{W'}]$ be another choice. By the same procedure as in the proof of Lemma 4.22, we deform $g_W(1)$ and deform-and-reverse $g_{W'}(1)$ to get $g_W(1)_{\text{def}}$ and $g_{W'}(1)_{\text{def},\text{rev}}$ on W and $W'$ respectively, so that they glue together along M. We denote the resulting element by $[W \cup_M W', g_W \cup_{W'}, t_W \cup_{W'}, f_W \cup_{W'}] \in C_{\Delta,n-1}^G(X)$. By Claim 4.27, we have + +$$ (4.105) \qquad \langle \mathrm{cw}_{g_W}(f_W^*\alpha), t_W \rangle - \langle \mathrm{cw}_{g_{W'}}(f_{W'}^*\alpha), t_{W'} \rangle = \langle \mathrm{cw}_{g_{W \cup W'}}(f_{W \cup W'}^*\alpha), t_{W \cup W'} \rangle. $$ + +Moreover, by (4.100) we have + +$$ (4.106) \qquad h([W \cup_M W', g_W \cup_{W'}, t_W \cup_{W'}, f_W \cup_{W'}]) = \langle \mathrm{cw}_{g_{W \cup W'}}(f_{W \cup W'}^*\alpha), t_{W \cup W'} \rangle. $$ +---PAGE_BREAK--- + +Finally we have + +$$ +\begin{align*} +(4.107) \quad & h([W, g_W, t_W, f_W]) - h([W', g_{W'}, t_{W'}, f_{W'}]) \\ +&= h([W, g_W(1)_{\text{def}}, t_W, f_W]) + h([W', g_{W'}(1)_{\text{def,rev}}, t_{W'}, f_{W'}]) \\ +&= h([W \cup_M W', g_{W\cup W'}(t_W \cup_{W'} f_W), f_{W\cup W'}]). +\end{align*} +$$ + +The first equality in (4.107) follows from Lemma 4.61. The second equality in (4.107) follows from the gluing formula in Proposition 4.92. Combining (4.105), (4.106) and (4.107) we get the desired well-definedness. + +For the compatibility, take any geometric singular stable tangential $G$-cycle $(M, g_M, t_M, f_M)$ of dimension $(n-2)$ over $A$ and its bordism data $(W, g_W, t_W, f_W)$. The compatibility condition is equivalent to the following. + +$$ +(4.108) \qquad h_A([M, g_M, t_M, f_M]) = \langle \mathrm{cw}_{g_W}(f_W^*\alpha|_A), t_W \rangle \pmod{\mathbb{Z}}. +$$ + +In this case we can take this ($W, g_W, t_W, f_W$) in (4.104), and by Lemma 4.91 we have $h([W, g_W, t_W, f_W]) = 0$. Thus (4.104) implies (4.108), and the compatibility follows. + +From the above argument and (4.102), we see that the compatibility con- +dition in Definition 4.56 (1) (c) with $\alpha|_A$ and (4.104) already determines the +well-defined homomorphism $h_A$ on the subgroup of $C_{\Delta,n-2}^G(A)$ generated by +the image of $\partial: C_{\Delta,n-1}^G(X,A) \to C_{\Delta,n-2}^G(A)$ and the kernel of the forgetful +map $C_{\Delta,n-2}^G(A) \to \Omega_{n-2}^G(A)$. Since $\mathbb{R}/\mathbb{Z}$ is an injective group, there exists +a group homomorphism $h_A: C_{\Delta,n-2}^G(A) \to \mathbb{R}/\mathbb{Z}$ extending it. It satisfies +the desired properties and completes the proof of the exactness of (4.95) at +$(I\Omega_{\text{sing}}^G)^n(X,A)$. + +Finally we prove the exactness of (4.95) at $(I\Omega_{\text{sing}}^G)^n(X)$. Suppose an element $I_{\text{sing}}((\lambda, h)) \in (I\Omega_{\text{sing}}^G)^n(X)$ satisfies $i^* \circ I_{\text{sing}}((\lambda, h)) = 0$. Then there exists an element $\alpha \in C^{n-1}(A; V_{I\Omega_G}^\bullet)$ with $a_{\text{sing}}(\alpha) = (\lambda|_A, h|_{C_{\Delta,n-1}^G(A)})$ in $(I\widehat{\Omega_{\text{sing}}^G})^n(A)$. This means that we have + +$$ +(4.109) \qquad \delta\alpha = \lambda|_A \text{ in } C^n(A; V_{I\Omega_G}^\bullet), +$$ + +$$ +(4.110) \qquad h = \operatorname{cw}(\alpha) \text{ on } C_{\Delta,n-1}^G(A). +$$ + +We have $\lambda - \delta\bar{\alpha} \in Z^n(X, A; V_{I\Omega_G}^\bullet)$ by (4.109). We construct an element in $(I\widehat{\Omega_{\text{sing}}^G})^n(X, A)$ of the form $(\lambda - \delta\bar{\alpha}, h')$ such that + +$$ +(4.111) \qquad h' = h - \mathrm{cw}(\bar{\alpha}) \text{ on } C_{\Delta, n-1}^{G}(X) +$$ + +This will imply that $I_{\text{sing}}((\lambda, h))$ is in the image of $j^*$. First we check the +following integrality condition, + +$$ +(4.112) \qquad \mathrm{cw}(\lambda - \delta\bar{\alpha}) \in \operatorname{Hom}(\Omega_n^G(X, A), \mathbb{Z}). +$$ + +Take any geometric singular stable tangential $G$-cycle $(W, g, t, f)$ of dimension $n$ over $(X, A)$. The compatibility condition in Definition 4.56 (1) (c) for $(\lambda, h) \in (I\widehat{\Omega_{\text{sing}}^G})^n(X)$, applied to $(M, g_M, t_M, f_M) = \partial(W, g, t, f)$ and $(W, g_W, t_W, f_W) = (W, g, t, f)$, implies that + +$$ +(4.113) \qquad h(\partial[W, g, t, f]) = \langle cw_g(f^*\lambda), t \rangle \pmod{\mathbb{Z}}. +$$ +---PAGE_BREAK--- + +But we have $\partial[W, g, t, f] \in C_{\Delta, n-1}^G(A)$, so (4.110) and (4.113) combine to give + +$$ \langle \mathrm{cw}_{\partial g}((f|_{\partial W})^{*\bar{\alpha}}), \partial t \rangle = \langle \mathrm{cw}_g(f^*\lambda), t \rangle \pmod{\mathbb{Z}}. $$ + +This implies (4.112). + +On the subgroup $C_{\Delta,n-1}^G(X)$ of $C_{\Delta,n-1}^G(X, A)$, we define the value of $h'$ by (4.111). Then it satisfies the compatibility condition with $\lambda - \delta\bar{\alpha}$ in Definition 4.56, because we have the compatibility condition for $(\lambda, h)$. From this and (4.112), the requirement (4.111) and the compatibility condition already determines the well-defined homomorphism $h'$ on the subgroup of $C_{\Delta,n-1}^G(X, A)$ generated by $C_{\Delta,n-1}^G(X)$ and the kernel of the forgetful map $C_{\Delta,n-1}^G(X, A) \to \Omega_{n-1}^G(X, A)$. Since $\mathbb{R}/\mathbb{Z}$ is an injective group, there exists a group homomorphism $h': C_{\Delta,n-1}^G(X, A) \to \mathbb{R}/\mathbb{Z}$ extending it. It satisfies the desired properties and completes the proof of the exactness of (4.95) at $(I\Omega_{\mathrm{sing}}^G)^n(X)$. This finishes the proof of (2) and completes the proof of Proposition 4.94. $\square$ + +We have defined the natural transformation $\int: (I\Omega_{\mathrm{sing}}^G)^{*+1}(S^1 \times -) \to (I\Omega_{\mathrm{sing}}^G)^*(-)$ in Definition 4.64. We now show that it coincides with the $S^1$-integration map for $\{(I\Omega_{\mathrm{sing}}^G)^*, \delta^*\}$ given in (2.9). + +**Proposition 4.114.** The natural transformation $\int : (I\Omega_{\mathrm{sing}}^G)^{*+1}(S^1 \times -) \to (I\Omega_{\mathrm{sing}}^G)^*(-)$ in Definition 4.64 coincides with the $S^1$-integration map for $\{(I\Omega_{\mathrm{sing}}^G)^*, \delta^*\}$ given in (2.9). + +*Proof.* In this proof, we denote by $\int'$ the transformations defined in Definition 4.64, in order to distinguish it from the true $S^1$-integration map defined in (2.9). We use the following notations for the standard maps, + +$$ i: (D^1, S^0) \to (S^1, \{\text{pt}\}), $$ + +$$ j: S^1 \to (S^1, \{\text{pt}\}). $$ + +It is enough to show the following. + +(i) For any CW-pair $(X, A)$, the following composition is the identity. + +(4.115) + +$$ (I\Omega_{\mathrm{sing}}^G)^n(X, A) \xrightarrow{\mathrm{susp}} (I\Omega_{\mathrm{sing}}^G)^{n+1}((S^1, \{\mathrm{pt}\}) \times (X, A)) \xrightarrow{(j\times id)^*} (I\Omega_{\mathrm{sing}}^G)^{n+1}(S^1 \times (X, A)) $$ + +$$ \xrightarrow{\int'} (I\Omega_{\mathrm{sing}}^G)^n(X, A). $$ + +Here the first map is defined in (2.8). + +(ii) For any CW-pair $(X, A)$, the following composition is zero. + +(4.116) + +$$ (I\Omega_{\mathrm{sing}}^G)^{n+1}(X, A) \xrightarrow{(k\times\mathrm{id})^*} (I\Omega_{\mathrm{sing}}^G)^{n+1}(S^1 \times (X, A)) \xrightarrow{\int'} (I\Omega_{\mathrm{sing}}^G)^n(X, A). $$ + +Here $k: \{\mathrm{pt}\} \to S^1$ is the inclusion of the basepoint. + +The item (ii) is easy, using the bordism data $(D^2 \times M, g_{D^2}, f \circ \mathrm{pr}_M)$ for $[S^1 \times M, g_{S^1}, P_{S^1}t, p_{S^1} \times f] \in C_{\Delta,n}^G(S^1 \times (X,A))$ (where $p_{S^1}: S^1 \to \{\mathrm{pt}\}$ and +---PAGE_BREAK--- + +$g_{S^1}$ is defined in Definition 4.33). The proof is left to the reader. In the following we prove (i). We set + +$$ \operatorname{desusp}' := \int' \circ (j \times \mathrm{id})^* : (\widehat{\Omega}_{\mathrm{sing}}^G)^{n+1}((S^1, \{\mathrm{pt}\}) \times (X, A)) \to (\widehat{\Omega}_{\mathrm{sing}}^G)^n(X, A) $$ + +$$ \widehat{\operatorname{desusp}}' := \int' \circ (j \times \mathrm{id})^* : (\widehat{\Omega}_{\mathrm{sing}}^G)^{n+1}((S^1, \{\mathrm{pt}\}) \times (X, A)) \to (\widehat{\Omega}_{\mathrm{sing}}^G)^n(X, A), $$ + +where $\int'$ for $(\widehat{\Omega}_{\mathrm{sing}}^G)^*$ is also defined in Definition 4.64. + +For a CW-pair $(X, A)$, we can always use $(X, A) \to (X/A, \{\mathrm{pt}\})$ which induces isomorphisms on generalized cohomology theories by the excision property. Thus we only consider a pair $(X, \{\mathrm{pt}\})$. + +For a pair $(X, \{\mathrm{pt}\})$, in any generalized cohomology theory $E$ we have a canonically split short exact sequence + +$$ 0 \to E^n(X, \{\mathrm{pt}\}) \to E^n(X) \to E^n(\{\mathrm{pt}\}) \to 0. $$ + +This follows from the long exact sequence for the pair $(X, \{\mathrm{pt}\})$ and the fact that there is a trivial retraction $X \to \{\mathrm{pt}\}$. Therefore, we can regard $E^n(X, \{\mathrm{pt}\})$ as a subgroup of $E^n(X)$. + +We denote the reduced cone of a pointed space $(X, \{\mathrm{pt}\})$ as $\widetilde{\mathrm{cone}}(X) := ([-1, 1] \times X)/(\{-1\} \times X \cup [-1, 1] \times \{\mathrm{pt}\})$. Also we denote $X^+ = X \cup \{\mathrm{pt}\}$. From the map $(X \times D^1, X \times S^0, X \times \{-1\}) \to (\widetilde{\mathrm{cone}}(X), X, \{\mathrm{pt}\})$ we get a commutative diagram + +where the composition of two arrows in each row is the suspension for $(X, \{\mathrm{pt}\})$ and $(X^+, \{\mathrm{pt}\})$, respectively. We have used $E^n(X) = E^n(X \times S^0, X \times \{-1\})$, and the coboundary map $\delta$ in the bottom row in (4.117) is given by regarding $X = \{1\} \times X \subset D^1 \times X$ (More precisely it is the coboundary map composed with the inclusion $E^n(X) \hookrightarrow E^n(S^0 \times X)$, but we just denote this by $\delta$). + +For the $S^1$-integration map we also have a commutative diagram which follows from the naturality of $\int'$, + +$$ +\begin{tikzcd} +(\widehat{\Omega}_{\text{sing}}^{G})^{n+1}(S^1 \times (X, \{\text{pt}\})) \arrow[r, "f'] & (\widehat{\Omega}_{\text{sing}}^{G})^{n}(X, \{\text{pt}\}) \\ +(\widehat{\Omega}_{\text{sing}}^{G})^{n+1}(S^1 \times (X^{+}, \{\text{pt}\})) \arrow[u,l, "f"] & (\widehat{\Omega}_{\text{sing}}^{G})^{n}(X) +\end{tikzcd} +$$ +---PAGE_BREAK--- + +Combining the suspension and $\int'$, we get the commutative diagram + +$$ +\begin{tikzcd} +& & (I\Omega_{\text{sing}}^G)^n(X, \{\text{pt}\}) \arrow[r, "susp"] & (I\Omega_{\text{sing}}^G)^{n+1}(\Sigma X, \{\text{pt}\}) \arrow[r, "desusp'] & (I\Omega_{\text{sing}}^G)^n(X, \{\text{pt}\}) \\ +& & \int & \arrow[uur]^{(I\Omega_{\text{sing}}^G)^n(X)} \arrow[dl]^{susp} & \int \arrow[ur]^{(I\Omega_{\text{sing}}^G)^{n+1}(\Sigma X^+, \{\text{pt}\})} \\ +& & & \int & (I\Omega_{\text{sing}}^G)^{n+1}(\Sigma X^+, \{\text{pt}\}) \arrow[r, "desusp'] & (I\Omega_{\text{sing}}^G)^n(X) +\end{tikzcd} +$$ + +Therefore, to show that the composition of the suspension and the desus- +pension is the identity, we only need to show it in the absolute case (X, ∅). + +Now our purpose is to show that the composition + +$$ +(4.118) \qquad (I\Omega_{\text{sing}}^G)^n(X) \xrightarrow{\delta} (I\Omega_{\text{sing}}^G)^{n+1}((D^1, S^0) \times X) \underset{\sim}{\overset{(i\times id)^*}{\leftarrow}} (I\Omega_{\text{sing}}^G)^{n+1}((S^1, \{\text{pt}\}) \times X) \\ +\qquad \longrightarrow (I\Omega_{\text{sing}}^G)^n(X) +$$ + +is the identity. + +Let us take $(\lambda, h) \in (\widetilde{I\Omega}_{\mathrm{sing}}^{G})^{n}(X)$ and take an element $(\Lambda', H') \in (\widetilde{I\Omega}_{\mathrm{sing}}^{G})^{n+1}((S^{1}, \{\mathrm{pt}\}) \times X)$ whose equivalence class $I(\Lambda', H') \in (\widetilde{I\Omega}_{\mathrm{sing}}^{G})^{n+1}((S^{1}, \{\mathrm{pt}\}) \times X)$ is obtained from the equivalence class $I(\lambda, h)$ under the first and second maps in (4.118). + +For a technical reason (see footnote 12) we replace the representative +($\Lambda', H'$) by the following procedure. Let us take a smooth map $\chi: (D^1, S^0) \to +(D^1, S^0)$ such that it is homotopic to $\mathrm{id}_{D^1}$, $\chi([-1, -0.5]) = \{-1\}$ and +$\chi([0.5, 1]) = \{1\}$. By abuse of notation we also denote by $\chi$ the induced map +$\chi: (S^1, \{\mathrm{pt}\}) \to (S^1, \{\mathrm{pt}\})$. We put $(\Lambda, H) := (\chi \times \mathrm{id}_X)^*(\Lambda', H')$. Since $(\chi \times \mathrm{id}_X)^*$ is identity on $(I\Omega_{\mathrm{sing}}^G)^{n+1}((S^1, \{\mathrm{pt}\}) \times X)$, $(\Lambda, H) \in (\widetilde{I\Omega}_{\mathrm{sing}}^G)^{n+1}((S^1, \{\mathrm{pt}\}) \times X)$ also satisfies the desired property. Our goal is to show that $\mathrm{desusp}'(\Lambda, H)-(\lambda, h) \in \mathrm{Im}(a_{\mathrm{sing}})$. + +We evaluate $\widehat{\mathrm{desusp}}'(\Lambda, H) \in (\widehat{I\Omega}_{\mathrm{sing}}^{G})^{n}(X)$ on $(M, g, t, f)$. By definition of $\widehat{\mathrm{desusp}}'$, it is given by the evaluation of $-(j \times \mathrm{id})^*(\Lambda, H) = -((\chi \circ j) \times \mathrm{id})^*(\Lambda', H')$ on $(S^1 \times M, g_{S^1}, P_{S^1}t$, $(\mathrm{id}_{S^1} \times f))$. This is the same as the evaluation of $-(\Lambda', H')$ on $(S^1 \times M, g_{S^1}, P_{S^1}t$, $(\chi \circ j) \times f)$. + +Let us notice that $\{\text{pt}\} \times M \subset S^1 \times M$ is mapped to $\{\text{pt}\} \times X \subset S^1 \times X$. +Using the gluing formula (Proposition 4.92), Remark 4.34 and Lemma 4.61, +we see that the result is unchanged if we evaluate $-(\Lambda', H')$ on¹² + +$$ +(D^1 \times M, g_{D1}, P_{D1}t, (i \circ \chi) \times f), +$$ + +where $g_{D1}$ is defined by (3.3). Thus its evaluation is the same as that of the pullback + +$$ +-(i \times \operatorname{id})^*(\Lambda', H') \in (\widehat{\mathbb{I}\Omega}_{\text{sing}}^{G})^{n+1}((D^1, S^0) \times X) +$$ + +on $(D^1 \times M, g_{D1}, P_{D1}t, \chi \times f)$. By definition of $(\Lambda', H')$, there exists an +element $\alpha' \in C^n((D^1, S^0) \times X; V_{I\Omega^G}^\bullet)$ such that + +$$ +(4.119) \qquad (i \times \mathrm{id})^*(\Lambda', H') = \hat{\delta}(\lambda, h) + a_{\mathrm{sing}}(\alpha'). +$$ + +12The reason for using (Λ, H) instead of (Λ', H') is that idD1 is not compatible with +the collar structure on D1, so the gluing formula cannot be applied here. +---PAGE_BREAK--- + +Notice that $D^1 \times M$ has two boundaries $\{-1\} \times M$ and $\{1\} \times M$ which are mapped to $\{-1\} \times X$ and $\{1\} \times X$, respectively. By using the definition of $\tilde{\delta}$, one can check that the evaluation of $\tilde{\delta}(\lambda, h)$ on $(D^1 \times M, g_{D^1}, P_{D^1}t, \chi \times f)$ is precisely given by $-h(M, g, t, f)$. This check requires + +$$ \langle \mathrm{cw}_{g_{D^1}} ((\chi \times f)^*\bar{\lambda}), P_{D^1}t \rangle = 0 $$ + +which can be shown by a similar computation as in the proof of Claim 4.120 below by using the definition of the prism operator $P_{D^1}$. + +On the other hand, the evaluation of $a_{\text{sing}}(\alpha')$ is given as follows. Set $\alpha := (\chi \times \text{id})^*\alpha'$. We denote by $P_{D^1}^* : C^n(D^1 \times X; V_{I^{\Omega G}}^\bullet) \to C^{n-1}(X; V_{I^{\Omega G}}^\bullet)$ the dual of $P_{D^1}$. We claim that + +**Claim 4.120.** Let $\alpha' \in C^n((D^1, S^0) \times X; V_{I^{\Omega G}}^\bullet)$ be any element and set $\alpha := (\chi \times \text{id})^*\alpha'$. The result of the evaluation of $a_{\text{sing}}(\alpha')$ on $(D^1 \times M, g_{D^1}, P_{D^1}t, \chi \times f)$ is the same as that of the evaluation of $a_{\text{sing}}(P_{D^1}^*\alpha)$ on $(M, g, t, f)$. + +The proof of Claim 4.120 is given below. Admitting the claim, we obtain + +$$ (4.121) \qquad \widehat{\operatorname{desusp}}'(\Lambda, H) = (\lambda, h) - a_{\operatorname{sing}}(P_{D^1}^*\alpha). $$ + +Indeed, the coincidence of the $\mathbb{R}/\mathbb{Z}$-valued homomorphism part follows from the above argument, and the coincidence of the cocycle part is obtained by applying $P_{D^1}^* \circ (\chi \times \text{id})^*$ to the both sides of the equality + +$$ (i \times \text{id})^* \Lambda' = \delta \bar{\lambda} + \delta \alpha' $$ + +which follows from (4.119) (note that $P_{D^1}^*(\delta\alpha) = -\delta(P_{D^1}^*\alpha)$ which follows from the formula (2.52)). This completes the proof. $\square$ + +*Proof of Claim 4.120.* Let $g = (d, P, \nabla, \psi)$. It is enough to show in the case where the image of $\alpha'$ in $C^n((D^1, S^0) \times X; V_{I^{\Omega G_d}}^\bullet)$ is of the form $\alpha'_d = \xi \otimes \phi$ for $\xi \in C^m((D^1, S^0) \times X; \mathbb{R})$ and $\phi \in V_{I^{\Omega G_d}}^{n-m}$. Then + +$$ \mathrm{cw}_{g_{D^1}}((\chi \times f)^*\alpha') = (\chi \times f)^*\xi \cup \mathrm{pr}_M^* \mathrm{cw}_g(\phi). $$ + +Let $\sigma: \Delta^{n-1} \to M$ be a simplex and $\tilde{\sigma}: \Delta^{n-1} \to \mathrm{Ori}(M)$ be its parallel lift. Recall the definition of the Prism operator (2.48). Using the same notation for vertices as there, we have + +$$ +\begin{aligned} +& \langle \mathrm{cw}_{g_{D^1}}((\chi \times f)^*\alpha'), P_{D^1} \tilde{\sigma} \rangle_{D^1 \times M} \\ +& = \left( \sum_{i=0}^{m-1} (-1)^i \langle (\chi \times \mathrm{id}_X)^*\xi, (\mathrm{id}_{D^1} \times f_*\sigma)|[v_0, \dots, v_i, w_i, \dots, w_{m-1}] \rangle_{D^1 \times M} \right) \\ +& \qquad \cdot \langle \mathrm{cw}_g(\phi), \tilde{\sigma}|[u_{m-1}, \dots, u_{n-1}] \rangle_M. +\end{aligned} +$$ + +where we have used the fact that $\langle (\chi \times \mathrm{id}_X)^*\xi, (\mathrm{id}_{D^1} \times f_*\sigma)|[v_0, \dots, v_m] \rangle = 0$ since $\xi$ is trivial on $\{0, 1\} \times X$. By the definition of $P_{D^1}$, this is equal to + +$$ +\begin{align*} +& \langle P_{D^1}^*(\chi \times \mathrm{id}_X)^*\xi, f_*\sigma|[u_0, \dots, u_{m-1}] \rangle_M \cdot \langle \mathrm{cw}_g(\phi), \tilde{\sigma}|[u_{m-1}, \dots, u_{n-1}] \rangle_M \\ +& = \langle f^*P_{D^1}^*(\chi \times \mathrm{id}_X)^*\xi \cup \mathrm{cw}_g(\phi), \tilde{\sigma} \rangle_M \\ +& = \langle \mathrm{cw}_g(f^*P_{D^1}^*\alpha), \tilde{\sigma} \rangle_M. +\end{align*} +$$ + +We conclude that + +$$ +\langle \mathrm{cw}_{g_{D^1}}((\chi \times f)^*\alpha'), P_{D^1}t \rangle_{D^1\times M} = \langle \mathrm{cw}_g(f^*P_{D^1}^*\alpha), t\rangle_M. +$$ +---PAGE_BREAK--- + +This is the desired result. + +□ + +**Proposition 4.122.** The quintuple $((\widehat{I\Omega^G_{dR}})^*, R, I, a, \int)$ is a differential extension of $\{(I\Omega^G_{sing})^*, \delta^*\}$ with $S^1$-integration (Definition 2.57 and Definition 2.62). + +*Proof.* The fact that the quadruple $((\widehat{I\Omega^G_{dR}})^*, R, I, a)$ is a differential extension follows from Proposition 4.30. For the $S^1$-integration, we check the commutativity of the middle square in (2.63). The other conditions are easily checked. By Proposition 4.114, it is enough to show that the induced homomorphism $\int : (I\Omega^G_{dR})^{*+1}(S^1 \times -) \to (I\Omega^G_{dR})^*(-)$ defined in (4.36) coincides with $\int : (I\Omega^G_{sing})^{*+1}(S^1 \times -) \to (I\Omega^G_{sing})^*(-)$ defined in (4.65) under the isomorphism $(I\Omega^G_{dR})^* \simeq (I\Omega^G_{sing})^*$ on manifolds. + +Recall that, in Proposition 4.70 and its proof, the isomorphism $(I\Omega^G_{sing})^* \simeq (I\Omega^G_{dR})^*$ on manifolds is given by passing to smooth singular model and constructing homomorphisms $(\widehat{I\Omega^G_{dR}})^* \to (\widehat{I\Omega^G_{sing,\infty}})^*$ and $(\widehat{I\Omega^G_{sing}})^* \to (\widehat{I\Omega^G_{sing,\infty}})^*$ which induces the isomorphisms on the cohomology level. The latter is just the forgetful homomorphism, but the former used a choice of natural cochain homotopy $B$ connecting $\wedge$ on differential forms and $\cup$ on smooth singular cochains (2.44). So from now on we fix such a natural cochain homotopy $B$. + +Passing from the singular model to the smooth singular model is also easy here. We can also define the smooth singular version of the $S^1$-integration map in Definition 4.64 in the obvious way. It is compatible with the singular version under the forgetful map. Now it is enough to show the commutativity of the following diagram for any manifold $X$. + +$$ +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] +(4.123) \quad & (\widehat{I\Omega^G_{dR}})^{n+1}(S^1 \times X) \arrow[r] & (\widehat{I\Omega^G_{sing,\infty}})^{n+1}(S^1 \times X) \\ +& \bigvee \arrow[u,r, "\int^{dR}"] \arrow[d, "\int^{\text{sing},\infty}"'] & \\ +& (\widehat{I\Omega^G_{dR}})^n(X) \arrow[r] & (\widehat{I\Omega^G_{sing,\infty}})^n(X). +\end{tikzcd} +$$ + +Here, in this proof we denote objects in differential models by the symbol "dR" and smooth singular models by "sing, ∞". But the $S^1$-integration map for differential forms (2.60) and smooth singular cochains (2.53) are compatible with the inclusion $\Omega^* \hookrightarrow C^\infty$, so we just denote them by the same symbol $\int$. + +Take $(\omega, h_{dR}) \in (\widehat{I\Omega^G_{dR}})^{n+1}(S^1 \times X)$. Recalling the definition of the trans- formation $(\widehat{I\Omega^G_{dR}})^* \to (\widehat{I\Omega^G_{sing,\infty}})^*$ given in the proof of Proposition 4.70, it maps to $(\omega, h_{sing,\infty}) \in (\widehat{I\Omega^G_{sing,\infty}})^{n+1}(S^1 \times X)$ under the refinement of the top horizontal arrow in (4.123), where, using the notations in that proof, + +$$ +(4.124) \quad h_{\mathrm{sing},\infty}([N,g,t,f]) := h_{\mathrm{dR}}([N,g,f]) - \langle B_g(f^*\omega), t\rangle_N. +$$ + +It maps to + +$$ +(4.125) \quad \left( \int \omega, \int_{-\infty}^{\infty} h_{\text{sing},\infty} \right) = \left( \int \omega, \int_{-\infty}^{\infty} h_{\text{sing},\infty} \right) \in (\widehat{I\Omega^G_{\text{sing},\infty}})^n(X) +$$ +---PAGE_BREAK--- + +under the refinement of the right vertical map in (4.123). We have + +$$ +\begin{align*} +& \left( \int^{\mathrm{sing},\infty} h_{\mathrm{sing},\infty} \right) ([M, g, t, f]) \\ +&= -h_{\mathrm{sing},\infty}([S^1 \times M, g_{S^1}, P_{S^1}t, \mathrm{id}_{S^1} \times f]) \\ +&= -h_{\mathrm{dR}}([S^1 \times M, g_{S^1}, \mathrm{id}_{S^1} \times f]) + \langle B_{g_{S^1}}((\mathrm{id}_{S^1} \times f)^*\omega), P_{S^1}t \rangle_{S^1 \times M} \\ +&= -h_{\mathrm{dR}}([S^1 \times M, g_{S^1}, \mathrm{id}_{S^1} \times f]) + \left\langle \int B_{g_{S^1}}((\mathrm{id}_{S^1} \times f)^*\omega), t \right\rangle_M . +\end{align*} +$$ + +Here in the last equation we use the identification $\text{Ori}(S^1 \times M) \simeq \text{pr}_M^*\text{Ori}(M)$ using $T(S^1 \times M) = \mathbb{R} \oplus \text{pr}_M^*TM$ and the convention (2.35) and (2.36), as explained in the last paragraph of Subsection 2.4. The $S^1$-integration on cochains is extended to cochains with coefficients in orientation bundles in the obvious way. + +On the other hand, under the other composition, $(\omega, h_{dR})$ maps to + +$$ +(4.127) \quad \left( \int \omega, \left( \int^{dR} h_{dR} \right)_{\text{sing},\infty} \right) = \left( \int \omega, \left( \int^{dR} h_{dR} \right)_{\text{sing},\infty} \right) \in (\widehat{I\Omega}_{\text{sing},\infty}^G)^n(X), +$$ + +where + +$$ +(4.128) \quad \left( \int^{dR} h_{dR} \right)_{\text{sing},\infty} ([M,g,t,f]) = \left( \int^{dR} h_{dR} \right) ([M,g,f]) - \left\langle B_g \left( f^* \int \omega \right), t \right\rangle_M \\ += -h_{dR}([S^1 \times M, g_{S^1}, \text{id}_{S^1} \times f]) - \left\langle B_g \left( f^* \int \omega \right), t \right\rangle_M +$$ + +To see that the two elements (4.125) and (4.127) coincide, by (4.126) and (4.128), it is enough to show that, for any $[M, g, f] \in C_{\infty, n-1}^G(X)$, the following element in $C_\infty^{n-1}(M; \text{Ori}(M))$, + +$$ +(4.129) \quad \int B_{g_{S^1}}((\mathrm{id}_{S^1} \times f)^*\omega) + B_g(f^*\int\omega), +$$ + +is a coboundary. + +Let $g = (d, P, \nabla, \psi)$. It is enough to show in the case where the image of $\omega$ in $\Omega_{\mathrm{clo}}^{n+1}(S^1 \times X; V_{I\Omega^{G_d}}^\bullet)$ is of the form $\omega_d = \xi \otimes \phi$ for $\xi \in \Omega_{\mathrm{clo}}^m(S^1 \times X)$ and $\phi \in V_{I\Omega^{G_d}}^{n+1-m}$. Then by the definition of $B_g$ in (4.74) and (4.76) and the definition of $g_{S^1}$, we have + +$$ +(4.130) \qquad B_g \left( f^* \int \omega \right) = B \left( f^* \int \xi, cw_g(\phi) \right) = B \left( \int (\mathrm{id}_{S^1} \times f)^* \xi, cw_g(\phi) \right), +$$ + +$$ +B_{g_{S^1}}((\mathrm{id}_{S^1} \times f)^*\omega) = B((\mathrm{id}_{S^1} \times f)^*\xi, \mathrm{pr}_M^*cw_g(\phi)). +$$ + +Here the second equality uses $\mathrm{Ori}(S^1 \times M) \simeq \mathrm{pr}_M^*\mathrm{Ori}(M)$ again. Now the desired result follows by the following claim. + +**Claim 4.131.** Let $B$ be a natural cochain homotopy between $\wedge$ and $\cup$ as in (2.45), (2.46). Let $M$ be a manifold and $E$ be a local coefficient system of +---PAGE_BREAK--- + +$\mathbb{R}$-modules on $M$. Then, for any $a \in \Omega_{\text{clo}}^{r+1}(S^1 \times M)$ and $b \in \Omega_{\text{clo}}^s(M; E)$, +the following smooth singular cochain in $C_{\infty}^{r+s}(M; E)$, + +$$ +\int B (a, \mathrm{pr}_M^* b) + B \left( \int a, b \right), +$$ + +is a coboundary. + +Proof of Claim 4.131. Fix $\tau_{S^1} \in \Omega_{\text{clo}}^1(S^1)$ representing the fundamental class for the standard orientation. Any $a \in \Omega_{\text{clo}}^{r+1}(S^1 \times M)$ can be decomposed as, + +$$ +(4.132) \qquad a = \tau_{S^1} \wedge \mathrm{pr}_M^* c + \mathrm{pr}_M^* e + df, +$$ + +for some $c \in \Omega_{\text{clo}}^r(M)$, $e \in \Omega_{\text{clo}}^{r+1}(M)$ and $f \in \Omega^r(S^1 \times M)$. Then, for the second term in (4.132), first we have + +$$ +\int \mathrm{pr}_M^* e = 0, +$$ + +because $\int \circ \mathrm{pr}_M^* = 0$ on differential forms. Using the naturality of $B$ and +$\int \circ \mathrm{pr}_M^* = Q^*\delta - \delta Q^*$ in the notation of (2.54), we have + +$$ +\begin{align*} +\int B (\mathrm{pr}_M^* e, \mathrm{pr}_M^* b) + B \left( \int \mathrm{pr}_M^* e, b \right) +&= \int \mathrm{pr}_M^* B(e, b) \\ +&\equiv Q^* \delta B(e, b) \mod (\mathrm{Im}\delta) \\ +&= Q^* (e \wedge b - e \cup b) \\ +&= 0, +\end{align*} +$$ + +where the third equality used (2.45), and the fourth equality used the fact +that $Q^*$ vanishes on differential forms and also cochains of the form $\omega \cup \eta$ +for two differential forms, which is the consequence of the formula (2.51) for +$Q$. + +For the third term in (4.132), applying (2.45), we have + +$$ +f \wedge \mathrm{pr}_M^* b - f \cup \mathrm{pr}_M^* b = \delta B(f, \mathrm{pr}_M^* b) + B(df, \mathrm{pr}_M^* b), +$$ + +$$ +\left(\int f\right) \wedge b - \left(\int f\right) \cup b = \delta B\left(\int f, b\right) + B\left(d\int f, b\right). +$$ + +Using $d \circ \int = - \int d \circ \int$, $\delta \circ \int = - \int \delta$ and (2.55), we get + +$$ +\begin{align*} +&\int B(df, \mathrm{pr}_M^* b) + B\left(\int df, b\right) \\ +&\equiv \int (f \wedge \mathrm{pr}_M^* b - f \cup \mathrm{pr}_M^* b) - \left(\left(\int f\right) \wedge b - \left(\int f\right) \cup b\right) \mod (\mathrm{Im}\delta) \\ +&= 0. +\end{align*} +$$ + +Thus, we are left to prove that the following cochain is a coboundary. + +$$ +(4.133) \qquad \int B (\tau_{S^1} \wedge \mathrm{pr}_M^* c, \mathrm{pr}_M^* b) + B(c, b). +$$ + +This follows from the following consideration. Given a homotopy $B$ as above +and fixing $\tau_{S^1}$, we can consider another natural cochain homotopy $B'$ defined +by, on each manifold $M$, + +$$ +B'(\omega, \eta) := - \int B (\mathrm{pr}_{S^1}^* \tau_{S^1} \wedge \mathrm{pr}_M^* \omega, \mathrm{pr}_M^* \eta). \\ +$$ +---PAGE_BREAK--- + +It is natural in $M$, and satisfies (2.45) because, + +$$ +\begin{align*} +& \delta B'(\omega, \eta) \\ +&= \int \delta B (\mathrm{pr}_{S1}^* \tau_{S1} \wedge \mathrm{pr}_M^* \omega, \mathrm{pr}_M^* \eta) \\ +&= \int \left( (\mathrm{pr}_{S1}^* \tau_{S1} \wedge \mathrm{pr}_M^* \omega \wedge \mathrm{pr}_M^* \eta - (\mathrm{pr}_{S1}^* \tau_{S1} \wedge \mathrm{pr}_M^* \omega) \cup \mathrm{pr}_M^* \eta) \right. \\ +&\qquad \left. + \int \left( B(\mathrm{pr}_{S1}^* \tau_{S1} \wedge \mathrm{pr}_M^* d\omega, \eta) + (-1)^{|\omega|} B(\mathrm{pr}_{S1}^* \tau_{S1} \wedge \mathrm{pr}_M^* \omega, d\eta) \right) \right) \\ +&= \omega \wedge \eta - \omega \cup \eta - B'(d\omega, \eta) - (-1)^{|\omega|} B'(\omega, d\eta). +\end{align*} +$$ + +Here the first equality used $\delta \circ \int = - \int \circ \delta$ (2.54) and the second used the formula (2.45) for $B$. Thus $B'$ is indeed another natural cochain homotopy between $\wedge$ and $\cup$. + +Since any two choices of such natural cochain homotopies are naturally +cochain homotopic, in particular for any manifold $M$ and for any closed $c$ +and $b$, the cochain + +$$ +B(c, b) - B'(c, b) +$$ + +is a coboundary. This implies that (4.133) is a coboundary, and completes +the proof. + +Applying Claim 4.131 to (4.130), we see that (4.129) is a coboundary, and +the proof of Proposition 4.122 is complete. + +**4.4. Self-duality homomorphisms.** In this subsection, we relate our mod- +els *IZdR* and *IZsing* with the ordinary cohomology theory. Recall that we +have the self-duality element $\gamma \in [*HZ, *IZ]*$ for the ordinary cohomology, +defined in (2.17). By Lemma 2.18, we have the commutative diagram of +generalized cohomology theories, + +$$ +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] +H^{*-1}(-; \mathbb{R}/\mathbb{Z}) \arrow[r, "{\delta_H}"'] & H^*(-; \mathbb{Z}) \arrow[r, "i_H"] & H^*(-; \mathbb{R}) \\ +\bigvee_{\substack{\gamma \\ |\gamma| = 1}} & \gamma & i \\ +\operatorname{Hom}(\pi_{-1}^{st}(-), \mathbb{R}/\mathbb{Z}) \arrow[r, "{\delta}"] & *[0]![-1]{-};\mathbb{Z} \arrow[r, "i"] & H^*(-; \mathbb{R}). +\end{tikzcd} +$$ + +We are going to construct natural transformations $\gamma_{\text{sing}} : H^*(-; \mathbb{Z}) \to (IZ_{\text{sing}})^*(-)$, +$\gamma_{\text{dR}} : H^*(-; \mathbb{Z}) \to (IZ_{\text{dR}})^*(-)$ and $\hat{\gamma}_{\text{dR}} : \hat{H}(-; \mathbb{Z}) \to (\hat{IZ}_{\text{dR}})^*(-)$ which fit +into the corresponding commutative diagram (Lemma 4.135 and Proposition 4.137). In Subsection 5.1 we are going to show that these natural +transformations correspond to, and refine, the self-duality element $\gamma$ under +the isomorphism $IZ \simeq IZ_{\text{sing}} \simeq IZ_{\text{dR}}$ proved there. + +For the singular model, the definition is simply the following. + +**Definition 4.134** ($\gamma_{\text{sing}}$). Let $(X, A)$ be a pair of topological spaces and $n \in \mathbb{Z}$. We define the homomorphism + +$$ +\gamma_{\text{sing}}^{n}: H_{\text{sing}}^{n}(X, A; \mathbb{Z}) \rightarrow (I\mathbb{Z}_{\text{sing}})^{n}(X, A), +$$ + +by sending the equivalence class of $\lambda \in Z^n(X, A; \mathbb{Z})$ to the class $[(\lambda_R, 0)]$. +Here $\lambda_R \in Z^n(X, A; \mathbb{R})$ is the $\mathbb{R}$-reduction of the element $\lambda$, and the pair +---PAGE_BREAK--- + +$(\lambda_{\mathbb{R}}, 0)$ satisfies the compatibility condition in Definition 4.56 (1) (c) because $\lambda_{\mathbb{R}}$ actually takes values in $\mathbb{Z}$. + +We easily see that $\gamma_{\text{sing}}^*$ is functorial and commute with the coboundary maps for ordinary cohomology and for $IZ_{\text{sing}}^*$. Thus it gives a morphism of generalized cohomology theories denoted by $\gamma_{\text{sing}}: HZ^* \to IZ_{\text{sing}}^*$. + +When $(X, A)$ is a pair of manifolds, we also define + +$$ \gamma_{\text{sing}}^n : H_{\text{sing},\infty}^n(X, A; \mathbb{Z}) \to (I\mathbb{Z}_{\text{sing},\infty})^n(X, A), $$ + +by the same way. Obviously it induces the same transformation as $\gamma_{\text{sing}}$ under the isomorphism $H_{\text{sing}}^* \simeq H_{\text{sing},\infty}^*$ on manifolds. + +We easily get the following lemma. + +**Lemma 4.135.** We have the commutative diagram of generalized cohomology theories, + +$$ +\begin{tikzcd} +H^{*-1}(-; \mathbb{R}/\mathbb{Z}) \arrow[r, "δ_H"] & H^*(-; \mathbb{Z}) \arrow[r, "ι_H"] & H^*(-; \mathbb{R}) \\ +\hom(\pi_{*-1}^{\text{st}}(-), \mathbb{R}/\mathbb{Z}) \arrow[u, "γ_{\mathbb{R}/\mathbb{Z}}"] \arrow[r, "p_{\text{sing}}"] & (I\mathbb{Z}_{\text{sing}})^*(-) \arrow[u, "γ_{\text{sing}}"] \arrow[r, "ch_{\text{sing}}"] & H^*(-; \mathbb{R}). \\ +\hline +\hom(\pi_{*-1}^{\text{st}}(-), \mathbb{R}/\mathbb{Z}) \arrow[u, "γ_{\mathbb{R}/\mathbb{Z}}"] \arrow[r, "p_{\text{sing}}"] & (I\mathbb{Z}_{\text{sing}})^*(-) \arrow[u, "γ_{\text{sing}}"] \arrow[r, "ch_{\text{sing}}"] & H^*(-; \mathbb{R}). +\end{tikzcd} +$$ + +Now we turn to the differential model. To define the homomorphism $\hat{\gamma}_{dR}: \hat{H}^*(-; \mathbb{Z}) \to (\hat{I}\mathbb{Z}_{dR})^*(-)$ from the ordinary differential cohomology theory, we use the Cheeger-Simons model for the ordinary differential cohomology explained in Example 2.58. In order to define $\hat{\gamma}_{dR}$, we remark the following. Let $(ω, k) ∈ \hat{H}_{CS}^n(X, A; \mathbb{Z})$. Then we get a group homomorphism also denoted by the same symbol $k$, + +$$ k: C_{\infty,n-1}^{\mathrm{fr}}(X,A) \to \mathbb{R}/\mathbb{Z} $$ + +by, given a geometric smooth stable tangential fr-cycle $(M, g, f)$ of dimension $(n-1)$ over $(X, A)$, choosing any representative $t_M \in Z_{\infty,n-1}(X, A; \mathbb{Z})$ of the fundamental class and applying $k$ to $t_M$. This value does not depend on the choice of $t_M$ because of the compatibility condition for $(ω, k)$. + +**Definition 4.136** ($\hat{\gamma}_{dR}$ and $\gamma_{dR}$). For a pair of manifolds $(X, A)$ and $n \in \mathbb{Z}$, we define a homomorphism + +$$ \hat{\gamma}_{dR}^n : \hat{H}_{CS}^n(X, A; \mathbb{Z}) \to (\widehat{I\mathbb{Z}_{dR}})^n(X, A) $$ + +by sending an element $(ω, k)$ to $(ω, k)$. The compatibility condition (Definition 4.11 (1) (c)) for the pair $(ω, k)$ follows from the compatibility condition (2.59) for the pair $(ω, k)$. + +We easily see that we have $a_{dR} = \hat{\gamma}_{dR}^n \circ a_{CS}$, so it induces the homomorphism on the quotient, + +$$ \gamma_{dR}^n : H^n(X, A; \mathbb{Z}) \to (I\mathbb{Z}_{dR})^n(X, A). $$ + +We also easily see that these homomorphisms are functorial, so gives natural transformations $\hat{\gamma}_{dR}: \hat{H}^*(-; \mathbb{Z}) \to I\mathbb{Z}_{dR}^*$ and $\gamma_{dR}: HZ^* \to I\mathbb{Z}_{dR}^*$ between functors $MfdPair^{\mathrm{op}} \to Ab^\mathbb{Z}$. + +The transformations $\gamma_{\mathrm{sing}}$ and $\gamma_{dR}$ coincide on manifolds, as follows. +---PAGE_BREAK--- + +**Proposition 4.137.** The composition of $\gamma_{dR}$ and the natural isomorphism $IZ_{dR}^* \simeq IZ_{sing,\infty}^* \simeq IZ_{sing}^*$ in Proposition 4.70 is equal to $\gamma_{sing}$. Moreover, $\hat{\gamma}_{dR}$ is a natural transformation between generalized differential cohomology theories refining $\gamma_{sing}$. + +*Proof.* As we did in the proof of Proposition 4.122, we need to recall the construction of the transformation $(\widehat{IZ}_{dR})^* \to (\widehat{IZ}_{sing,\infty})^*$ which induces the isomorphism on the level of cohomology. It was constructed in the proof of Proposition 4.70, and in general we needed to choose a natural cochain homotopy between $\wedge$ and $\cup$. However, actually in the case of $IZ$ it does not depend on that choice, and the construction there simply gives + +$$ (\widehat{IZ}_{dR})^n(X, A) \to (\widehat{IZ}_{sing,\infty})^n(X, A), \quad (\omega, h) \mapsto (\omega, h \circ \text{fgt}), $$ + +where fgt: $C_{\Delta,\infty,n-1}^{fr}(X,A) \to C_{\infty,n-1}^{fr}(X,A)$ is the forgetful map. This is because the Chern-Weil construction in the case $G=\text{fr} = \{\text{1}\}$ is trivial, and in particular we have $cw^{sing} = cw^{dR}$. + +Thus, it is enough to check that, given an element $(\omega, k) \in \widehat{H}_{CS}^n(X, A; \mathbb{Z})$ and $\lambda \in Z_\infty^n(X, A; \mathbb{Z})$ which represents the same class in $H_{\text{sing},\infty}^n(X, A; \mathbb{Z})$, we have + +$$ I_{\text{sing},\infty}((\omega, k \circ \text{fgt})) = I_{\text{sing},\infty}((\lambda_{\mathbb{R}}(0))). $$ + +Since we know that $(\omega, k)$ and $\lambda$ represents the same cohomology class, there exists an element $\alpha \in C_\infty^{n-1}(X, A; \mathbb{R})$ such that + +$$ \begin{aligned} \omega - \lambda_{\mathbb{R}} &= \delta\alpha, \\ k &= \alpha \pmod{\mathbb{Z}}. \end{aligned} $$ + +Using it we have $(\omega, k \circ \text{fgt}) - (\lambda_{\mathbb{R}}, 0) = a_{\text{sing},\infty}(\alpha)$. This completes the proof. $\square$ + +## 5. THE PROOF OF THE MAIN RESULTS + +In this section we prove the main result of this paper. We prove that the generalized cohomology theory $(I\Omega_{\text{sing}}^G)^*$ constructed in Section 4 is indeed isomorphic to the Anderson dual to the $G$-bordism theory $(I\Omega^G)^*$ (Theorem 5.21). We first prove the isomorphism $IZ_{\text{sing}}^* \simeq IZ^*$ in Subsection 5.1, which corresponds to the case $G = \text{fr} = \{\text{1}\}$. Using it we prove the isomorphism for general $G$ in Subsection 5.2. + +Recall that the term “generalized cohomology theory” means generalized cohomology theory for *CW-pairs* (Remark 2.6). + +**5.1. The proof of $IZ_{\text{sing}}^* \simeq IZ^*$.** The goal of this subsection is to prove the following result. + +**Theorem 5.1.** (1) There exists a unique isomorphism of generalized co-homology theories, + +$$ \varphi_{\text{sing}} : IZ_{\text{sing}}^* \simeq IZ^*, $$ +---PAGE_BREAK--- + +such that the following diagrams of generalized cohomology theories commute. + +(2) Moreover, we can refine the statement (1) in the category $\operatorname{Ho(Sp)}$ as follows. Fix an $\Omega$-spectrum $IZ_{\text{sing}}$ representing the cohomology theory $\{IZ_{\text{sing}}^*, \delta^*\}$. + +(a) The lifts of $\mathrm{ch}_{\mathrm{sing}}: IZ_{\mathrm{sing}}^* \to \mathbb{H}\mathbb{R}^*$ and $\gamma_{\mathrm{sing}}: HZ^* \to IZ_{\mathrm{sing}}^*$ to elements $\mathrm{ch}_{\mathrm{sing}} \in [IZ_{\mathrm{sing}}, \mathbb{H}\mathbb{R}]$ and $\gamma_{\mathrm{sing}} \in [HZ, IZ_{\mathrm{sing}}]$ are unique, respectively. + +(b) There exists a unique isomorphism $\varphi_{\mathrm{sing}} \in [IZ_{\mathrm{sing}}, IZ]$ in $\operatorname{Ho(Sp)}$ +such that, taking an arbitrary lift $p_{\mathrm{sing}} \in [\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), IZ_{\mathrm{sing}}]$ of +$p_{\mathrm{sing}} : \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})^* \to IZ_{\mathrm{sing}}^*$, the following diagrams in $\operatorname{Ho(Sp)}$ +commute. + +The rest of this subsection is devoted to the proof of Theorem 5.1. Ac- +tually, we do not have to go back to the explicit construction of $IZ_{\text{sing}}^*$ anymore. We know that the generalized cohomology theory $\{IZ_{\text{sing}}^*, \delta^*\}$ has the following properties. + +* We have morphisms of generalized cohomology theories $p_{\text{sing}} : \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})^* \to IZ_{\text{sing}}^*$, $\text{ch}_{\text{sing}} : IZ_{\text{sing}}^* \to \mathbb{H}\mathbb{R}^*$ (Definition 4.60) and $\gamma_{\text{sing}} : HZ^* \to IZ_{\text{sing}}^*$ (Definition 4.134). + +* For any $n \in \mathbb{Z}$ and any CW pair $(X, A)$, the following sequence is exact (Proposition 4.67). + +$$ +(5.2) \qquad H^{n-1}(X, A; \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\pi_{n-1}^{\text{st}}(X, A), \mathbb{R}/\mathbb{Z}) \xrightarrow{p_{\text{sing}}} IZ_{\text{sing}}^n(X, A) \\ +\qquad \qquad \xrightarrow{\text{ch}_{\text{sing}}} H^n(X, A; \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\pi_n^{\text{st}}(X, A), \mathbb{R}/\mathbb{Z}). +$$ + +* The following diagram of generalized cohomology theories commutes (Lemma 4.135). + +We also know that *IZ* has the same properties. We are going to show that these properties characterizes *IZ* (Proposition 5.3 and Proposition 5.11). Theorem 5.1 follows from them. +---PAGE_BREAK--- + +**Proposition 5.3.** Let $\mathcal{I}^*$ be a generalized cohomology theory. Let $p': \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})^* \to \mathcal{I}^*$, $l': \mathcal{I}^* \to \mathbb{H}\mathbb{R}^*$, and $\gamma': \mathbb{H}\mathbb{Z}^* \to \mathcal{I}^*$ be three morphisms of generalized cohomology theories. Assume the following conditions. + +(i) For any $n \in \mathbb{Z}$ and any CW pair $(X, A)$, the following sequence is exact. + +$$ (5.4) \qquad H^{n-1}(X, A; \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\pi_{n-1}^{\mathrm{st}}(X, A), \mathbb{R}/\mathbb{Z}) \xrightarrow{p'} \mathcal{I}^n(X, A) $$ + +$$ \xrightarrow{l'} H^n(X, A; \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\pi_n^{\mathrm{st}}(X, A), \mathbb{R}/\mathbb{Z}). $$ + +(ii) The following diagram of generalized cohomology theories commutes. + +Then, there exist a unique isomorphism $\varphi: \mathcal{I}^* \to \mathcal{IZ}^*$ of generalized cohomology theories such that the following diagrams of generalized cohomology theories commute. + +To show Theorem 5.1 (2), we need to pass to the category Ho(Sp). Lemma 5.6 below allows us to lift the data in Proposition 5.3 to the corresponding data in Ho(Sp). Using it, we are going to show Proposition 5.11, which is the statement in Ho(Sp). From that proposition, Proposition 5.3 follows directly. + +**Lemma 5.6.** Assume that a generalized cohomology theory $\mathcal{I}^*$ satisfies the conditions in Proposition 5.3. Fix an $\Omega$-spectrum $\mathcal{I} = \{\mathcal{I}_n, \sigma_n^{\mathcal{I}}\}_{n \in \mathbb{Z}}$ representing $\mathcal{I}^*$. + +(1) The lifts of $l': \mathcal{I}^* \to \mathbb{H}\mathbb{R}^*$ and $\gamma': \mathbb{H}\mathbb{Z}^* \to \mathcal{I}^*$ to elements in $[\mathcal{I}, \mathbb{H}\mathbb{R}]$ and $[\mathbb{H}\mathbb{Z}, \mathcal{I}]$ are unique, respectively. We denote them by $l' \in [\mathcal{I}, \mathbb{H}\mathbb{R}]$ and $\gamma' \in [\mathbb{H}\mathbb{Z}, \mathcal{I}]$. + +(2) Take an arbitrary lift $p' \in [\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), \mathcal{I}]$ of $p': \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})^* \to \mathcal{I}^*$. Then the following diagram in Ho($\mathbf{Sp}$) commutes. + +$$ (5.7) \qquad \begin{tikzcd}[column sep=2.8em, row sep=2.8em] + \Sigma^{-1}H(\mathbb{R}/\mathbb{Z}) & \xleftarrow{\mathrm{pH}} & \mathbb{H}\mathbb{Z} & \xleftarrow{\mathrm{lH}} & \mathbb{H}\mathbb{R} \\ + \big|_{\gamma_{\mathbb{R}/\mathbb{Z}}} & & \big|_{\gamma'} & & \\ + \Sigma^{-1}I(\mathbb{R}/\mathbb{Z}) & \xleftarrow{\mathrm{p'}} & \mathcal{I} & \xleftarrow{\mathrm{l'}} & \mathbb{H}\mathbb{R}. +\end{tikzcd} $$ + +Proof. For (1), by Fact 2.4, it is enough to prove the following. + +$$ (5.8) \qquad \lim_{\underset{n}{\longrightarrow}} \tilde{\mathcal{H}}^{n-1}(\mathcal{I}_n; \mathbb{R}) = 0 $$ + +$$ (5.9) \qquad \lim_{\underset{n}{\longleftarrow}} \tilde{\mathcal{T}}^{n-1}(K(\mathbb{Z}, n)) = 0. $$ +---PAGE_BREAK--- + +For (5.8), note that we have $\tilde{H}^{n-1}(\mathcal{I}_n; \mathbb{R}) = \text{Hom}(\pi_{n-1}^{\text{st}}(\mathcal{I}_n), \mathbb{R})$. Since $\mathbb{R}$ is an injective group, by Lemma 2.5 we get (5.8). Next we prove (5.9). We apply the exact sequence (5.4) for $(X, A) = (K(\mathbb{Z}, n+1), \{\ast\})$. Since $\pi_{n-1}^{\text{st}}(K(\mathbb{Z}, n+1)) = \pi_{n-1}(K(\mathbb{Z}, n+1)) = 0$ and $\tilde{H}^n(K(\mathbb{Z}, n+1); \mathbb{R}) = \text{Hom}(\pi_n^{\text{st}}(K(\mathbb{Z}, n+1)), \mathbb{R}) = \text{Hom}(\pi_n(K(\mathbb{Z}, n+1)), \mathbb{R}) = 0$ by the Freudenthal suspension theorem, we get $\tilde{\mathcal{I}}^n(K(\mathbb{Z}, n+1)) = 0$ and (5.9) follows. + +Now we prove (2). We know that the diagram (5.7) induces the commutative diagram of generalized cohomology theories (5.5). By (1) we see that the right square in (5.7) commutes. To see the commutativity of the left square, by Lemma 2.5 it is enough to show that + +$$ (5.10) \qquad \lim_{\underset{n}{\longrightarrow}}^1 \tilde{\mathcal{I}}^n(K(\mathbb{R}/\mathbb{Z}, n)) = 0. $$ + +To prove this we apply the exact sequence (5.4) for $(X, A) = (K(\mathbb{R}/\mathbb{Z}, n), \{\ast\})$. Since $\pi_{n-1}^{\text{st}}(K(\mathbb{R}/\mathbb{Z}, n)) = \pi_{n-1}(K(\mathbb{R}/\mathbb{Z}, n)) = 0$ by the Freudenthal suspension theorem, we see that $\tilde{\mathcal{I}}^n(K(\mathbb{R}/\mathbb{Z}, n)) \simeq \text{Hom}(\pi_n^{\text{st}}(K(\mathbb{R}/\mathbb{Z}), n), \mathbb{Z})$. By this identification the homomorphism $\tilde{\mathcal{I}}^{n+1}(K(\mathbb{R}/\mathbb{Z}, n+1)) \to \tilde{\mathcal{I}}^n(K(\mathbb{R}/\mathbb{Z}, n))$ is induced by the homomorphism $\pi_n^{\text{st}}(K(\mathbb{R}/\mathbb{Z}, n)) \to \pi_{n+1}^{\text{st}}(K(\mathbb{R}/\mathbb{Z}, n+1))$ which is the identity on $\mathbb{R}/\mathbb{Z}$ by the Freudenthal suspension theorem for $n \ge 2$. Thus we get (5.10) and the result follows. □ + +From now on we take $\iota' \in [\mathcal{I}, H\mathbb{R}]$ and $\gamma' \in [HZ, \mathcal{I}]$ as in Lemma 5.6 (1) and fix an arbitrary lift $p' \in [\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), \mathcal{I}]$ of $p': \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})^* \to \mathcal{I}^*$. By Lemma 5.6 (2), they fit into the commutative diagram (5.7). + +**Proposition 5.11.** Assume that a generalized cohomology theory $\mathcal{I}^*$ satisfies the conditions in Proposition 5.3. Fix an $\Omega$-spectrum $\mathcal{I} = \{\mathcal{I}_n, \sigma_n^\mathcal{I}\}_{n \in \mathbb{Z}}$ representing $\mathcal{I}^*$. Take $\iota' \in [\mathcal{I}, H\mathbb{R}]$ and $\gamma' \in [HZ, \mathcal{I}]$ as in Lemma 5.6 (1) and fix an arbitrary lift $p' \in [\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), \mathcal{I}]$ of $p': \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})^* \to \mathcal{I}^*$. + +(1) There exists a unique element $\varphi \in [\mathcal{I}, \text{IZ}]$ such that the following diagram in $\text{Ho(Sp)}$ commutes. + +Moreover, the element $\varphi \in [\mathcal{I}, \text{IZ}]$ in (1) satisfies the following. + +(2) The following diagram in $\text{Ho(Sp)}$ commutes. + +(3) The following diagram in $\text{Ho(Sp)}$ commutes. +---PAGE_BREAK--- + +(4) $\varphi$ is an isomorphism in Ho(Sp). + +Obviously, Proposition 5.11 implies Proposition 5.3. As a preparation for the proof of Proposition 5.11, we compute the homotopy groups of $\mathcal{I}$. + +**Lemma 5.12.** Let $\mathcal{I}$ be a spectrum which represents a generalized cohomology theory $\mathcal{I}^*$ satisfying the conditions in Proposition 5.3. Then we have + +$$ \pi_n(\mathcal{I}) \simeq \begin{cases} 0 & (n \ge 1) \\ \mathbb{Z} & (n = 0) \\ 0 & (n = -1) \\ \operatorname{Hom}(\pi_{-n-1}^{\text{st}}(S^0), \mathbb{R}/\mathbb{Z}) & (n \le -2) \end{cases} $$ + +*Proof.* We have + +$$ \pi_n(\mathcal{I}) = \tilde{\mathcal{I}}^{-n}(S^0). $$ + +By the condition (i) of Proposition 5.3, the following sequence is exact. + +$$ +\begin{aligned} +\tilde{H}^{-n-1}(S^0; \mathbb{R}) &\xrightarrow{\pi} \operatorname{Hom}(\pi_{-n-1}^{\text{st}}(S^0), \mathbb{R}/\mathbb{Z}) \xrightarrow{p'} \tilde{\mathcal{I}}^{-n}(S^0) \\ +&\xrightarrow{\iota'} \tilde{H}^{-n}(S^0; \mathbb{R}) \xrightarrow{\pi} \operatorname{Hom}(\pi_{-n}^{\text{st}}(S^0), \mathbb{R}/\mathbb{Z}). +\end{aligned} +$$ + +From this we obtain the desired result. $\square$ + +*Proof of Proposition 5.11 (1).* By the exact sequence for the distinguished triangle (2.12), the following sequence is exact. + +$$ [\mathcal{I}, \Sigma^{-1} I(\mathbb{R}/\mathbb{Z})] \xrightarrow{p_*} [\mathcal{I}, \mathcal{I}\mathbb{Z}] \xrightarrow{\iota_*} [\mathcal{I}, H\mathbb{R}] \xrightarrow{\pi_*} [\mathcal{I}, I(\mathbb{R}/\mathbb{Z})]. $$ + +We have $[\mathcal{I}, \Sigma^{-1} I(\mathbb{R}/\mathbb{Z})] = \operatorname{Hom}(\pi_{-1}(\mathcal{I}), \mathbb{R}/\mathbb{Z}) = 0$ by Lemma 5.12. Thus it is enough to show that + +$$ (5.13) \qquad \pi \circ \iota' = 0 \in [\mathcal{I}, I(\mathbb{R}/\mathbb{Z})]. $$ + +By the condition (i) in Proposition 5.3, we see that $\pi \circ \iota' \in [\mathcal{I}, I(\mathbb{R}/\mathbb{Z})]$ induces the zero morphism $0: \mathcal{I}^* \to I(\mathbb{R}/\mathbb{Z})^*$ between generalized cohomology theories. By Fact 2.4, to get (5.13) it is enough to show that $\varinjlim_n I(\mathbb{R}/\mathbb{Z})^{n-1}(\mathcal{I}_n) = 0$. But this follows from $I(\mathbb{R}/\mathbb{Z})^{n-1}(\mathcal{I}_n) = \operatorname{Hom}(\pi_{n-1}^{\text{st}}(\mathcal{I}_n), \mathbb{R}/\mathbb{Z})$ and Lemma 2.5 since $\mathbb{R}/\mathbb{Z}$ is an injective abelian group. This completes the proof of Proposition 5.11 (1). $\square$ + +*Proof of Proposition 5.11 (2).* Consider the following diagram in Ho(Sp), + +$$ (5.14) $$ + +We have + +$$ (5.15) \qquad \iota_H = \iota' \circ \gamma', \quad \iota' = \iota \circ \varphi, \quad \iota_H = \iota \circ \gamma. $$ + +Indeed, the first equation follows from the commutativity of (5.7), the second follows from the definition of $\varphi$ and the third follows from the commutativity +---PAGE_BREAK--- + +of (2.19). By Lemma 2.20 and (5.15), we get the commutativity of the left +triangle of 5.14. This completes the proof of Proposition 5.11 (2). +□ + +*Proof of Proposition 5.11 (3).* Consider the following diagram in $Ho(\mathbf{Sp})$. + +$$ +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] + \Sigma^{-1}H(\mathbb{R}/\mathbb{Z}) \arrow[r, "$p_H$] & H\mathbb{Z} \arrow[d, "$\gamma_{\mathbb{R}/\mathbb{Z}}$] + \\ + \Sigma^{-1}I(\mathbb{R}/\mathbb{Z}) \arrow[r, "$p'$] & I \arrow[d, "$\gamma'$] + \\ + \mathbb{Z} \arrow[u, "$p$] & \mathbb{Z} +\end{tikzcd} +$$ + +What we want to show is the commutativity of the bottom triangle of (5.16). +We know that the right triangle is commutative by Proposition 5.11 (2). +Moreover, by Lemma 2.18 and the commutativity of (5.7), we have the +commutativity of the two squares, + +$$ +(5.17) \qquad p' \circ \gamma_{\mathbb{R}/\mathbb{Z}} = \gamma' \circ p_H, \quad p \circ \gamma_{\mathbb{R}/\mathbb{Z}} = \gamma \circ p_H. +$$ + +We need the following lemma. + +**Lemma 5.18.** The map + +$$ +\gamma_{\mathbb{R}/\mathbb{Z}}^* : [\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), I\mathbb{Z}] \to [\Sigma^{-1}H(\mathbb{R}/\mathbb{Z}), I\mathbb{Z}] +$$ + +is bijective. + +*Proof.* First remark that, since we have $\pi_1(I(\mathbb{R}/\mathbb{Z})) = [\mathbb{S}, \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})] = \text{Hom}(\pi_1([\mathbb{S}], \mathbb{R}/\mathbb{Z}) = 0)$ and $\pi_1(H(\mathbb{R}/\mathbb{Z})) = H^{-1}([\mathbb{S}]; \mathbb{R}/\mathbb{Z}) = 0$, + +$$ +[\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), H\mathbb{R}] = \operatorname{Hom}(\pi_1(I(\mathbb{R}/\mathbb{Z})), \mathbb{R}) = 0, +$$ + +$$ +[\Sigma^{-1} H(\mathbb{R}/\mathbb{Z}), H\mathbb{R}] = \operatorname{Hom}(\pi_1(H(\mathbb{R}/\mathbb{Z})), \mathbb{R}) = 0. +$$ + +Using this, by the exact sequence for the distinguished triangle (2.12) we +get the following commutative diagram, + +$$ +(5.19) \qquad +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] +[\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), \Sigma^{-1}H\mathbb{R}] \arrow[r, "-(\Sigma^{-1}\pi)_*"] & [\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})] \arrow[r, "p_*"] & [\Sigma^{-1}I(\mathbb{R}/\mathbb{Z}), I\mathbb{Z}] \arrow[r, "l_*"] & 0 +\\ +\bigdownarrow_{\gamma_{\mathbb{R}/\mathbb{Z}}^*} & \bigdownarrow_{\gamma_{\mathbb{R}/\mathbb{Z}}^*} & & \bigdownarrow_{\gamma_{\mathbb{R}/\mathbb{Z}}^*} +\\ +[\Sigma^{-1}H(\mathbb{R}/\mathbb{Z}), \Sigma^{-1}H\mathbb{R}] \arrow[r, "-(\Sigma^{-1}\pi)_*"] & [\Sigma^{-1}H(\mathbb{R}/\mathbb{Z}), \Sigma^{-1}I(\mathbb{R}/\mathbb{Z})] \arrow[r, "p_*"] & [\Sigma^{-1}H(\mathbb{R}/\mathbb{Z}), I\mathbb{Z}] \arrow[r, "l_*"] & 0 +\end{tikzcd} +$$ + +where the rows are exact. We want to show that the right vertical arrow +of (5.19) is an isomorphism. It is enough to show that the left and middle +vertical arrows are isomorphisms. These arrows are rewritten as + +$$ +\begin{align*} +\gamma_{\mathbb{R}/\mathbb{Z}}^* : \operatorname{Hom}(\pi_0(I(\mathbb{R}/\mathbb{Z})), \mathbb{R}) & \rightarrow \operatorname{Hom}(\pi_0(H(\mathbb{R}/\mathbb{Z})), \mathbb{R}), \\ +\gamma_{\mathbb{R}/\mathbb{Z}}^* : \operatorname{Hom}(\pi_0(I(\mathbb{R}/\mathbb{Z})), \mathbb{R}/\mathbb{Z}) & \rightarrow \operatorname{Hom}(\pi_0(H(\mathbb{R}/\mathbb{Z})), \mathbb{R}/\mathbb{Z}), +\end{align*} +$$ + +respectively. Thus it is enough to show that $\gamma_{\mathbb{R}/\mathbb{Z}} \in [H(\mathbb{R}/\mathbb{Z}), I(\mathbb{R}/\mathbb{Z})]$ induces the isomorphism on $\pi_0$, i.e., the homomorphism + +$$ +(5.20) \qquad (\gamma_{\mathbb{R}/\mathbb{Z}}^*) : [\mathbb{S}, H(\mathbb{R}/\mathbb{Z})] \to [\mathbb{S}, I(\mathbb{R}/\mathbb{Z})] +$$ +---PAGE_BREAK--- + +is an isomorphism. But this follows directly from the definition of $\gamma_{\mathbb{R}/Z}$. +Indeed, by Definition 2.17 the homomorphism (5.20) is induced by the +Hurewicz homomorphism for $\mathbb{S}$, + +$$ \text{Hur}^* : \operatorname{Hom}(H_0(\mathbb{S}; \mathbb{Z}), \mathbb{R}/\mathbb{Z}) \to \operatorname{Hom}(\pi_0(\mathbb{S}), \mathbb{R}/\mathbb{Z}) $$ + +Since Hur: $\pi_0(S) \to H_0(S; Z)$ is an isomorphism, we see that (5.20) is an +isomorphism. This completes the proof of Lemma 5.18. +□ + +By Proposition 5.11 (2), the equations (5.17) and Lemma 5.18, we get the +commutativity of the bottom triangle of (5.16). This completes the proof +of Proposition 5.11 (3). +□ + +*Proof of Proposition 5.11 (4) and Proposition 5.3.* To see that $\varphi$ is an isomorphism in $Ho(\mathbf{Sp})$, it is enough to show that it induces the isomorphisms on homotopy groups, i.e., that $\varphi_* : [\Sigma^n\mathbb{S}, \mathcal{I}] \to [\Sigma^n\mathbb{S}, \mathcal{IZ}]$ is an isomorphism for all $n \in \mathbb{Z}$. + +By Proposition 5.11 (1) and (3), the following diagram commutes. + +The first row is exact by the condition (i) in Proposition 5.3, and the second row is exact by (2.13). Thus by the five lemma we see that $\varphi_* : [\Sigma^n\mathbb{S}, \mathcal{I}] \to [\Sigma^n\mathbb{S}, \mathcal{IZ}]$ is an isomorphism for each $n \in \mathbb{Z}$. This completes the proof of Proposition 5.11 (4) and of Proposition 5.3. +$\square$ + +*Proof of Theorem 5.1.* As mentioned in the paragraph before Proposition 5.3, $\mathcal{IZ}_{\text{sing}}^*$ satisfies the conditions in Proposition 5.3. Theorem 5.1 (1) follows from Proposition 5.3, and (2) follows from Lemma 5.6 and Proposition 5.11. +$\square$ + +**5.2. The proof of $(I\Omega_{\text{sing}}^G)^* \simeq (I\Omega^G)^*$.** The goal of this subsection is to show the following result. + +**Theorem 5.21.** There exists an isomorphism of generalized cohomology theories, + +$$ (I\Omega_{\text{sing}}^{G})^{*} \simeq (I\Omega^{G})^{*}, $$ + +which makes the following diagram of generalized cohomology theories com- +mutative. + +As a preparation to the proof of Theorem 5.21, we discuss approximations +of the stable Madsen-Tillmann spectrum *MTG* by manifolds in Subsub- +section 5.2.1, which allows us to perform the geometric Pontryagin-Thom +construction explained in Subsection 3.2. Using it, the proof of Theorem +---PAGE_BREAK--- + +5.21 is given in Subsection 5.2.2. Subsection 5.2.3 is a supplemental material, where we explain a part of the construction in Subsection 5.2.2 in the differential model. + +5.2.1. *Approximations of the Madsen-Tillmann spectra by manifolds.* Now we discuss approximations of *MTG* by finite-dimensional manifolds. We use the notations in Subsection 2.3. Note that, in the construction of *MTG* there, although $\text{Gr}_d(d+q)$ is a manifold, $\mathcal{X}_d(d+q)$ is not in general. + +First we show that we can take a sequence of classifying spaces $\{BG_d, Bs_d\}_d$ which can be nicely approximated by manifolds. + +**Lemma 5.22.** We can choose $\{BG_d, Bs_d\}_d$ so that they have the following properties. + +(1) For each positive integer $p$, there is a subspace $B^p G_d \subset BG_d$ such that $B^p G_d$ is a closed manifold and $B^p G_d \hookrightarrow BG_d$ is $p$-connected. Moreover, $B^p G_d \subset B^{p+1} G_d$ and it is a smooth closed embedding. + +(2) The classifying map + +$$ \begin{array}{ccc} EG_d \times_{s_d} G_{d+1} & \xrightarrow{Es_d} & EG_{d+1} \\ \downarrow & & \downarrow \\ BG_d & \xrightarrow{Bs_d} & BG_{d+1} \end{array} $$ + +has the property that its restriction to $B^p G_d$ is a smooth map $B^p s_d: B^p G_d \to B^p G_{d+1}$. + +(3) $B^p s_d: B^p G_d \to B^p G_{d+1}$ is a smooth closed embedding and $Bs_d$ is a closed embedding. + +*Proof*. Inductively, suppose that we have constructed $BG_{d'}$ and $Bs_{d'-1}$ for $d' \le d-1$. We construct $BG_d$ and $Bs_{d-1}$. + +Fix a faithful representation of the compact group $G_d$, + +$$ \eta_d : G_d \to \text{O}(D_d, \mathbb{R}). $$ + +Inductively we take + +$$ N_{p,d} = p + (\dim B^p G_{d-1} + 1) + N_{p-1,d} $$ + +where we take $N_{-1,d} := 0$. Then we take $B^l G_d$ to be + +$$ B^l G_d := V_{D_d}(\mathbb{R}^{D_d+N_{p,d}})/\eta_d(G_d), $$ + +where $V_D(\mathbb{R}^{D+N})$ is the Stiefel manifold (the set of orthonormal $D$-frames in $\mathbb{R}^{D+N}$). Clearly we have a smooth closed embedding $B^l G_d \to B^{l+1} G_d$ which is at least $p$-connected. We set + +$$ B'G_d := \varinjlim_p B'^p G_d = V_{D_d}(\mathbb{R}^\infty)/\eta_d(G_d). $$ + +This is a classifying space for $G_d$ satisfying the condition (1) of the statement of the lemma. + +Consider a classifying map $BG_{d-1} \to B'G_d$ of the bundle $EG_{d-1} \times_{s_{d-1}} G_d$. Its restriction to $B^p G_{d-1}$ can be deformed to a map into $B'^p G_d$ which is unique up to homotopy, since we have taken $N_{p,d}$ so that $B^p G_d \to B'G_d$ is at least $(\dim B^p G_{d-1} + 1)$-connected. Inductively, we take $B^p G_{d-1} \to B'^p G_d$ to be smooth and extend it to $B^{p+1} G_{d-1} \to B'^{p+1} G_d$. In this way, we +---PAGE_BREAK--- + +obtain $B^p s_{d-1} : B^p G_{d-1} \to B'^p G_d$ and $B's_{d-1} : BG_{d-1} \to B'G_d$ with the +corresponding bundle maps which satisfy the property (2) of the statement +of the lemma. + +Finally, we modify the above construction so that (3) holds. We take +$M_{p,d} \in \mathbb{Z}$ to be + +$$ +M_{p,d} = p + 1 + (2\dim B^p G_{d-1}) + M_{p-1,d}. +$$ + +By Whitney embedding theorem we have a smooth embedding + +$$ +B^p G_{d-1} \to \mathbb{R}^{2\dim B^p G_{d-1}}. +$$ + +Suppose we already have $B^{p-1}G_{d-1} \to \mathbb{R}^{M_{p-1,d}}$. By embedding $B^p G_{d-1} \to \mathbb{R}^{2\dim B^p G_{d-1}}$ further in $\mathbb{R}^{M_{p,d}} = \mathbb{R}^{2\dim B^p G_{d-1}} \times \mathbb{R}^{p+1} \times \mathbb{R}^{M_{p-1,d}}$ and appropriately modifying it in the neighborhood of $B^{p-1}G_{d-1} \subset B^p G_{d-1}$, we get a smooth embedding such that the following diagram commutes; + +By the one-point compactification of $\mathbb{R}^{N_{p,d}}$ to $S^{N_{p,d}}$, we get + +$$ +\mathrm{emb}_{p,d-1}: B^p G_{d-1} \to S^{M_{p,d}}, +$$ + +such that it extends the previous embedding $\mathrm{emb}_{p-1,d-1} : B^{p-1}G_{d-1} \to S^{M_{p-1,d}} \subset S^{M_{p,d}}$. In this way we obtain an embedding + +$$ +\mathrm{emb}_{d-1} : BG_{d-1} \to S^{\infty} +$$ + +whose restriction to $B^p G_{d-1}$ is $\mathrm{emb}_{p,d-1}$. Now we set + +$$ +\begin{align*} +B^p G_d &:= B'^p G_d \times S^{M_{p,d}}, \\ +BG_d &:= B' G_d \times S^\infty = \varinjlim_p B^p G_d. +\end{align*} +$$ + +Notice that $B^p G_d \to BG_d$ still satisfies the condition (1) in the statement +of the lemma since $M_{p,d} > p$ and hence $S^{M_{p,d}}$ is $p$-connected. We also set + +$$ +\begin{align*} +B^p s_{d-1} &:= B'^p s_{d-1} \times \operatorname{emb}_{p,d-1}, \\ +Bs_{d-1} &:= B's_{d-1} \times \operatorname{emb}_{d-1}. +\end{align*} +$$ + +These $BG_d$ and $Bs_{d-1}$ satisfy all the conditions (1), (2) and (3) in the +statement of the lemma. This completes the proof. $\square$ + +Now we notice that if $BG_d$ is a classifying space for $G_d$, then + +$$ +(5.23) \qquad (EG_d \times V_d(\mathbb{R}^\infty))/G_d +$$ + +is also a classifying space for $G_d$, where $G_d$ acts on $V_d(\mathbb{R}^\infty)$ via $\rho_d : G_d \to O(d, \mathbb{R})$. We have the pullback diagram +---PAGE_BREAK--- + +Therefore, we explicitly define + +$$ +(5.24) \qquad \mathcal{X}_d(d+q) := (EG_d \times V_d(\mathbb{R}^{d+q}))/G_d. +$$ + +It has all the general properties required for $\mathcal{X}_d(d+q)$ in Subsection 2.3. + +By using the $BG_d$ appearing in Lemma 5.22, $\mathcal{X}_d(d+q)$ is realized as + +$$ +\mathcal{X}_d(d+q) = \varinjlim_p (E^p G_d \times V_d(\mathbb{R}^{d+q})) / G_d +$$ + +where $E^p G_d$ is the restriction of $E G_d$ to $B^p G_d$. Now each term $(E^p G_d \times V_d(\mathbb{R}^{d+q}))/G_d$ is a closed manifold. + +In the current model for $BG_d$, we have closed embeddings + +$$ +\mathcal{X}_d(d+q) \hookrightarrow \mathcal{X}_{d+1}(d+1+q). +$$ + +Thus we define + +$$ +\mathcal{X}(q) := \varinjlim_{d} \mathcal{X}_{d}(d+q) = \varinjlim_{d} \varinjlim_{p} (E^{p}G_{d} \times V_{d}(\mathbb{R}^{d+q}))/G_{d}. +$$ + +In terms of $\mathcal{X}(q)$ the $q$-th space of the Madsen-Tillmann spectrum is given +by + +$$ +(MTG)_q = \mathrm{Thom}(Q_q \to \mathcal{X}(q)) +$$ + +with the obvious structure map $\Sigma MTG_q \to MTG_{q+1}$. + +Now we define + +$$ +(5.25) \qquad \mathcal{X}_d^p(d+q) := (E^p G_d \times V_d(\mathbb{R}^{d+q}))/G_d, +$$ + +$$ +(5.26) \qquad \mathcal{X}^p(q) := \varinjlim_d (E^p G_d \times V_d(\mathbb{R}^{d+q}))/G_d +$$ + +Notice that $\mathcal{X}_d^p(d+q)$ is a closed manifold. The point is that if we are given any compact space $K$ and a map $f: K \to (MTG)_q$, its image is actually contained in + +$$ +\mathrm{Thom}(Q_q \to \mathcal{X}_d^p(d+q)) +$$ + +for some $d$ and $p$. + +The Thom space $\text{Thom}(Q_q \to \mathcal{X}^p(q))$ is an approximation of $(MTG)_q$ in +the following sense. + +**Proposition 5.27.** For each pair of positive integers $p$ and $q \ge 2$, the inclusion + +$$ +\operatorname{Thom}(Q_q \to \mathcal{X}^p(q)) \hookrightarrow (\operatorname{MTG})_q = \operatorname{Thom}(Q_q \to \mathcal{X}(q)) +$$ + +is $(p+q)$-connected. Moreover, composing it with the map $\Sigma^\infty(MTG)_q \to \Sigma^q MTG$, the induced map $w_q^p \in [\Sigma^\infty\operatorname{Thom}(Q_q \to \mathcal{X}^p(q)), \Sigma^q MTG]$ is $\min(p+q, 2q)$-connected. + +*Proof.* For each *d*, consider the following pullback diagram. + +$$ +\begin{array}{ccc} +\mathcal{X}_d^p(d+q) & \longrightarrow & \mathcal{X}_d(d+q). \\ +\downarrow & & \downarrow \\ +B^p G_d & \longrightarrow & BG_d +\end{array} +$$ + +Recall that, by Condition (1) in Lemma 5.22, the bottom arrow is *p*-connected. +Since the vertical arrows are fiber bundles, we see that the top arrow is also +*p*-connected. Passing to the direct limit in *d*, we also see that the inclusion +---PAGE_BREAK--- + +$\mathcal{X}(q) \to \mathcal{X}^p(q)$ is $p$-connected. Thus, the map on the Thom spaces of the rank-$q$ vector bundles $Q_q$ with $q \ge 2$ covering it is $(p+q)$-connected by [Rud98, Chapter IV, Lemma 5.20]. The last statement follows from Lemma 2.28. $\square$ + +One technical advantage of the above explicit construction of MTG, which will be used in the next subsection, is the following. + +(i) For each $p$, there exists a sequence $\{\nabla_d^p\}_d$ consisting of a $G_d$-connection $\nabla_d^p$ on $E^p G_d \to B^p G_d$ for each $d$, such that they are compatible with the bundle maps + +$$ +\begin{tikzcd} +& E^p G_d \times_{s_d} G_{d+1} \arrow[r, "E^p s_d"] & E^p G_{d+1} \arrow[d] \\ +B^p G_d \arrow[u] \arrow[r, "B^p s_d"] & B^p G_{d+1} \arrow[u] +\end{tikzcd} +$$ + +This is because $B^p s_d$ is a smooth closed embedding (Lemma 5.22 (3)). Such a sequence $\{\nabla_d^p\}_d$ also induces a $G_d$-connection on $E^p G_d \times V_d(\mathbb{R}^{d+q}) \to \mathcal{X}_d^p(d+q)$ for each $d$ and $q$ by pullback. They are compatible with the bundle maps covering $\mathcal{X}_d^p(d+q) \hookrightarrow \mathcal{X}_{d+1}^p(d+1+q)$ and $\mathcal{X}_d^p(d+q) \hookrightarrow \mathcal{X}_d^p(d+q+1)$. + +5.2.2. *The proof of Theorem 5.21.* Now we proceed to give a proof of Theorem 5.21. Recall that the spectrum representing $(I\Omega^G)^*$ is the function spectrum $F(MTG, IZ)$ (2.32), so we have + +$$ +(5.28) \qquad (I\Omega^G)^*(X, A) \simeq IZ^*((X/A) \wedge MTG) +$$ + +for any CW-pair $(X, A)$. The strategy of the proof of Theorem 5.21 is to construct an isomorphism corresponding to (5.28) in our models, and reduce to Theorem 5.1. + +First we note the following. + +**Lemma 5.29.** Let $n$ be a nonnegative integer. Assume that a continuous map $f: (X, A) \to (Y, B)$ between two pairs of spaces induces isomorphisms $f_*: \pi_j^{\text{st}}(X, A) \simeq \pi_j^{\text{st}}(Y, B)$ for $j = n-1, n$. Then we have + +$$ +f^* : \mathrm{IZ}^n(Y, B) \simeq \mathrm{IZ}^n(X, A). +$$ + +Proof. This follows from the exact sequence (2.13) and the five lemma. $\square$ + +Fix a nonnegative integer $n$. By Lemma 5.29 and Proposition 5.27, for $p \ge n+1$ and $q \ge n+1, 2$ we have an isomorphism (recall our convention (1.17)) + +$$ +(5.30) \qquad (\mathrm{id}_{X/A} \wedge w_q^p)^* \circ \mathrm{susp}^q : \mathrm{IZ}^n((X/A) \wedge \mathrm{MTG}) \simeq \mathrm{IZ}^{n+q}((X,A) \times \mathrm{Thom}_\infty(Q_q|\mathcal{X}^p(q))) +$$ + +for any CW-pair $(X, A)$. Combining it with (5.28), for $q \ge n+1, 2$ we get + +$$ +(5.31) \qquad (I\Omega^G)^n(X, A) \simeq I\mathbb{Z}^{n+q}((X, A) \times \operatorname{Thom}_\infty(Q_q | \mathcal{X}^p(q))). +$$ + +Now we are going to construct a natural transformation of generalized co-homology theories + +$$ +(5.32) \quad P_q^{*,p}: (I\Omega_{\text{sing}}^G)^*(-) \to (I\mathbb{Z}_{\text{sing}})^{*+q}(- \times \text{Thom}_{\infty}(Q_q|\mathcal{X}_p(q)), +$$ +---PAGE_BREAK--- + +for each pair of positive integers $p$ and $q$, which is going to realize the required isomorphism up to sign for $p \ge *+1$ and $q \ge *+1, 2$ (see the sign appearing in (5.79)). We first construct its refinement, which is a homomorphism between $\widehat{I\Omega_{\text{sing}}^G}$ and $\widehat{IZ}_{\text{sing}}$, using the geometric Pontryagin-Thom construction explained in Subsection 3.2. The idea of the construction is simple, but the details are complicated. We outline the construction of its differential version in Subsection 5.2.3 below. Since it is much simpler, we recommend the reader to take a look at it first. + +Fix positive integers $p$ and $q$. We fix the data (i) explained in the last part of Subsection 5.2.1. Recall that it consists of a sequence $\{\nabla_d^p\}_d$ which gives connections on the principal $G_d$-bundle on $\mathcal{X}_d^p(d+q)$ for each $d$, also denoted by $\nabla_d^p$. Moreover, we also fix the following. + +(ii) A representative of the Thom class $\tau_q \in C^q(\text{Thom}_\infty(Q_q|_{\mathcal{X}^p(q)}); \text{Ori}(\pi))$ for the vector bundle $\pi: Q_q \to \mathcal{X}^p(q)$. + +(iii) Note that we have a cochain map by restriction, + +$$ (5.33) \qquad C^\bullet(\mathcal{X}^p(q); \text{Ori}(\pi)) \to \varinjlim_d C_\infty^\bullet(\mathcal{X}_d^p(d+q); \text{Ori}(\pi)), $$ + +which is surjective and induces an isomorphism on the cohomology. +We fix a cochain map + +$$ s: \varprojlim_d C_\infty^\bullet(\mathcal{X}_d^p(d+q); \mathrm{Ori}(\pi)) \to C^\bullet(\mathcal{X}^p(q); \mathrm{Ori}(\pi)) $$ + +which is a splitting of (5.33). Such a map exists because (5.33) is +surjective quasi-isomorphism and $\mathbb{R}$ is injective (see [Qui67, Chapter +II, Section 4, item 5]). + +We define a natural transformation + +$$ \hat{\mathcal{P}}^{*,p}_{q, \nabla, \tau, s}: (\widehat{I\Omega_{\text{sing}}^G})^{*}(-) \to (\widehat{I\mathbb{Z}_{\text{sing}}})^{*+q}(- \times \text{Thom}_{\infty}(Q_q|\mathcal{X}^p(q))) $$ + +as follows (the subscript “$\nabla$, $\tau, s$” indicates the dependence on the chosen +data (i), (ii) and (iii) above). Fix an integer $n$ and a pair of topological +spaces $(X, A)$. Take $(\lambda, h) \in (\widehat{I\Omega_{\text{sing}}^G})^n(X, A)$. + +We construct $\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda) \in Z^{n+q}((X,A) \times \text{Thom}_{\infty}(Q_q|\mathcal{X}^p(q)); \mathbb{R})$. First we construct a cochain map + +$$ (5.34) \qquad \hat{\mathcal{P}}_{\nabla, \tau, s}: C^*(X, A; V_{I\Omega^G}^{\bullet}) \to C^{*+q}((X, A) \times \operatorname{Thom}_{\infty}(Q_q|\mathcal{X}^p(q)) ; \mathbb{R}). $$ + +Let us take $\mu \in C^m(X, A; V_{I\Omega^G}^\bullet)$. First fix $d \in \mathbb{Z}_{\ge 0}$. Apply the forgetful map $V_{I\Omega^G}^\bullet \to V_{I\Omega^G_d}^\bullet$ to the coefficient of $\lambda$ and denote by $\mu_d$ the resulting element. Write $\mu_d = \sum_j \xi_{d,j} \otimes \phi_{d,j}$ with $\xi_{d,j} \in C^*(X, A; \mathbb{R})$ and $\phi_{d,j} \in V_{I\Omega^G_d}^\bullet$. For each $j$ we apply the Chern-Weil construction (4.6) for $\phi_{d,j}$ with respect to $\nabla_d^p$ to get a form $cw\nabla_d^p(\phi_{d,j}) \in \Omega_{\text{clo}}^\bullet(\mathcal{X}_d^p(d+q); \text{Ori}(\pi))$. We regard it as a smooth singular cocycle. Using the splitting $s$ and $\text{Ori}(\pi) \simeq \text{Ori}(\pi)^*$, we get + +$$ (5.35) \qquad \sum_j \operatorname{pr}_X^* \xi_{d,j} + \operatorname{pr}_{D(Q_q)}^* (\pi^* s (\operatorname{cw}\nabla_d^p(\phi_{d,j})) + \tau_q) \in C^{m+q}((X,A) \times \operatorname{Thom}_\infty(Q_q|\mathcal{X}_d^{p(d+q)}); \mathbb{R}). $$ + +By the compatibility of the sequence $\{\nabla_d^p\}_d$, we see that (5.35) defines an element in $C^{m+q}((X, A) \times \text{Thom}_\infty(Q_q|\mathcal{X}^p(q)); \mathbb{R}) = \varinjlim_d C^{m+q}((X, A) \times +---PAGE_BREAK--- + +Thom$_\infty(Q_q|\mathcal{X}_d^p(d+q)); \mathbb{R}\rangle$. Thus we define the transformation (5.34) by + +$$ +(5.36) \quad \hat{\mathcal{P}}_{\nabla, \tau, s}(\mu) := \left\{ \sum_j \operatorname{pr}_X^* \xi_{d,j} \cup \operatorname{pr}_D^*(Q_q) \left( \pi^* s \left( \operatorname{cw}_{\nabla_d^p}(\phi_{d,j}) \right) \cup \tau_q \right) \right\}_d . +$$ + +Since $\pi^*s(\text{cw}_{\nabla_d^p}(\phi_{d,j})) \cup \tau_q$ is a cocycle, (5.34) is a cochain map. + +We apply the homomorphism (5.34) to $\lambda$ and get $\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda) \in Z^{n+q}((X,A) \times \text{Thom}_{\infty}(Q_q|\mathcal{X}^p(q)); \mathbb{R})$. We note that the associated homomorphism $\text{cw}(\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda)) \in \text{Hom}(\Omega_{n+q}^r((X,A) \times \text{Thom}_{\infty}(Q_q|\mathcal{X}^p(q))), \mathbb{R})$ is the pullback of the element $\text{ch}_{\text{sing}}(\lambda) = \text{cw}(\lambda) \in \text{Hom}(\Omega_n^G(X,A), \mathbb{R})$ under the homomorphism $\Omega_{n+q}^r((X,A) \times \text{Thom}_{\infty}(Q_q|\mathcal{X}^p(q))) \to \Omega_n^G(X,A)$, up to sign. In particular, by the integrality condition for $\lambda$ in Lemma 4.66, we get the integrality condition for $\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda)$, + +$$ +(5.37) \quad \mathrm{cw}(\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda)) \in \mathrm{Hom}(\Omega_{n+q}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}^{\mathrm{p}}(q))), \mathbb{Z}). +$$ + +Next we define a group homomorphism $\hat{\mathcal{P}}_{\nabla,\tau,s}(h) : C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}^p(q))) \to \mathbb{R}/\mathbb{Z}$. We are going to use the geometric Pontryagin-Thom construction in Subsection 3.2. + +Recall that we have defined the notion of “niceness with tubular neigh- +borhood structures” for elements in $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d+q))) +in Definition 3.15. Since we have + +$$ +\Omega_{n+q-1}^{\mathrm{fr}}((X, A) \times \mathrm{Thom}_{\infty}(Q_q | \mathcal{X}^p(q))) = \varinjlim_d \Omega_{n+q-1}^{\mathrm{fr}}((X, A) \times \mathrm{Thom}_{\infty}(Q_q | \mathcal{X}_d^p(d+q))) +$$ + +and any element in $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d+q)))$ is bordant to +a nice one, we see that the group $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_q^p))$ is +generated by + +(A) nice elements in $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d+q)))$ for each $d$, +and + +(B) the kernel of the forgetful homomorphism $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}^p(q))) \to$ +$\Omega_{n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}^p(q)))$. + +Thus, first we are going to define the value of $\hat{\mathcal{P}}_{\nabla,\tau,s}(h)$ for nice elements in +$C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d+q)))$ for each $d$. + +For a nice element $[M, g, t, f] \in C_{\Delta, n+q-1}^{\mathrm{fr}}((X, A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d+q)))$ with a tubular neighborhood structure $\pi_f$ for some $d$, we set + +$$ +(5.38) \quad \hat{\mathcal{P}}_{\nabla, \tau, s}(h)([M, g, t, f]) := h(\mathrm{PT}_{\nabla_d^p, \tau_q}[M, g, t, f]). +$$ + +Here + +$$ +\mathrm{PT}_{\nabla_d^p, \tau_q}[M, g, t, f] := [M_f, g_{f_*} \nabla_d^p, t_{M_f, \tau_q}, \mathrm{pr}_{(X,A)} \circ f|_{M_f}] \in C_{\Delta, n-1}^G(X, A) +$$ + +is defined in Definition 3.22. The well-definedness follows from Lemma 3.24 +(1) and Lemma 4.61. The above $[M,g,t,f]$ is also nice as an element in +$C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d'+q)))$ for $d' \ge d$. By the compatibility +of the sequence $\{\nabla_d^p\}_d$, using Lemma 3.24 (3) and Lemma 4.61, we see that +the value (5.38) does not depend on the choice of $d'$. +---PAGE_BREAK--- + +We need to check that the definition (5.38) satisfies the compatibility condition in Definition 4.56 (1) (c) with $\hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda)$ defined above. Take a nice element $[M, g_M, t_M, f_M] \in C_{\Delta, n+q-1}^{\text{fr}}((X, A) \times \text{Thom}_{\infty}(Q_q|\chi_d^p(d+q)))$ for some $d$ and fix a representative with a tubular neighborhood structure $\pi_{f_M}$, and take a bordism data $(W, g_W, t_W, f_W)$ of $(M, g_M, t_M, f_M)$. From now on, we trivialize $\text{Ori}(W)$ using the stable framing $g_W$. We need to check that (note that $\text{cw}_{g_W}(f_W^*\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda)) = f_W^*\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda)$ in this case) + +$$ (5.39) \quad \hat{\mathcal{P}}_{\nabla,\tau,s}(h)([M,g_M,t_M,f_M]) = \langle f_W^* \hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda), t_W \rangle_W \pmod{\mathbb{Z}}. $$ + +Replacing $d$ large enough if necessary, we may assume that the image of $f_W$ also lies in $X \times D(Q_q|\chi_d^p(d+q))$. First assume that $(W, g_W, t_W, f_W)$ is also nice with a tubular neighborhood structure $\pi_{f_W}$ extending $\pi_{f_M}$ (see Lemma 3.24 (4)). We denote by $\pi_{f_W}: \nu_f \to W_{f_W}$ the tubular neighborhood of $W_{f_W} = f_W^{-1}(X \times \mathfrak{x}_d^p(d+q))$ in the condition (B) in Definition 3.15. Then, the geometric Pontryagin-Thom construction in Definition 3.22 gives + +$$ \mathrm{PT}_{\nabla_d^p, \tau_q}(W, g_W, t_W, f_W) = (W_{f_W}, (g_W)_{f_W^* \nabla_d^p}, t_{W_{f_W}}, \tau_q, \mathrm{pr}_X \circ f_W |_{W_{f_W}}). $$ + +By Lemma 3.24 (4), $\mathrm{PT}_{\nabla_d^p, \tau_q} (W, g_W, t_W, f_W)$ is a bordism data for $\mathrm{PT}_{\nabla_d^p, \tau_q} (M, g_M, t_M, f_M)$. Thus the compatibility of $\lambda$ and $h$ implies + +$$ (5.40) \quad h([\mathrm{PT}_{\nabla_d^p, \tau_q} (M, g_M, t_M, f_M)]) = \left\langle \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}} ((\mathrm{pr}_X \circ f_W|_{W_{f_W}})^*\lambda), t_{W_{f_W}, \tau_q} \right\rangle_{W_{f_W}} \pmod{\mathbb{Z}}. $$ + +By (5.38), the left hand side of (5.40) coincides with the left hand side of (5.39). Next we check the equality + +$$ (5.41) \quad \langle f_W^* \hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda), t_W \rangle_W = \left\langle \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}} ((\mathrm{pr}_X \circ f_W|_{W_{f_W}})^*\lambda), t_{W_{f_W}, \tau_q} \right\rangle_{W_{f_W}}. $$ + +We may assume that $\lambda_d$ is of the form $\lambda_d = \xi \otimes \phi$ for $\xi \in Z^*(X, A; \mathbb{R})$ and $\phi \in V_{I\Omega^{G_d}}$. On $W$, we have + +$$ f_W^* \hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda) = f_W^* (\mathrm{pr}_X^\ast\xi \cup \mathrm{pr}_{D(Q_q}^\ast) (\pi^s(s (\mathrm{cw}_{\nabla_d^p}(\phi)) \cup \tau_q)). $$ + +Since we have $\nu_W = f_W^{-1}(X \times \mathring{D}(Q_q|\chi_d^p(d+q)))$, we have $f_W^*\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda) \in Z^{n+q}(W,W\setminus\nu_W;\mathbb{R})$. By Condition (C) in Definition 3.15, we have a decomposition $t_W = t_{\overline{\nu_W}} + (t_W - t_{\overline{\overline{\nu_W}}})$ with $t_{\overline{\overline{\nu_W}}} \in C_\infty^{n+q}(\overline{\nu_W};\mathbb{R})$ and $t_W - t_{\overline{\overline{\nu_W}}} \in C_\infty^{n+q}(W\setminus\nu_W;\mathbb{R})$. Thus we have + +$$ (5.42) \quad \langle f_W^* \hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda), t_W \rangle_W = \langle f_W^* \hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda), t_{\overline{\overline{\nu_W}}} \rangle_{\overline{\overline{\nu_W}}} $$ +---PAGE_BREAK--- + +The cocycle $f_W^* \hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda)$, regarded as a smooth singular cocycle on $(\bar{\nu}_W, \partial\nu_W)$, +can be written as + +$$ +(5.43) \qquad +\begin{aligned} +f_W^* \hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda) &= (\mathrm{pr}_X \circ f_W)^* \xi \cup \pi_{f_W}^* \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}(\phi) \cup (\mathrm{pr}_{D(Q_q)} \circ f_W)^* \tau_q \\ +&= \pi_{f_W}^* \left( (\mathrm{pr}_X \circ f_W |_{W_{f_W}})^* \xi \cup \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}(\phi) \right) \cup (\mathrm{pr}_{D(Q_q)} \circ f_W)^* \tau_q \\ +&\in Z_\infty^{n+q}(\bar{\nu}_W, \partial\nu_W; \mathbb{R}). +\end{aligned} +$$ + +The last equality used the commutativity of the diagram (3.16). On the +other hand, on $W_{f_W}$ we have + +$$ +(5.44) \qquad \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}((\mathrm{pr}_X \circ f_W|_{W_{f_W}})^*\lambda) = (\mathrm{pr}_X \circ f_W|_{W_{f_W}})^*\xi \cup \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}(\phi) \in Z_\infty^n(W_{f_W}; \mathbb{R}) +$$ + +We have + +$$ +\begin{align*} +& \langle f_W^* \hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda), t_W \rangle_W \\ +&= \langle f_W^* \hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda), t_{\overline{\nu_W}} \rangle_{\overline{\nu_W}} \\ +&= \left\langle \pi_{f_W}^* \left( (\mathrm{pr}_X \circ f_W |_{W_{f_W}})^* \xi \cup \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}(\phi) \right), (\mathrm{pr}_{D(Q_q)} \circ f_W)^* \tau_q \cap t_{\overline{\nu_W}} \right\rangle_{\overline{\nu_W}} \\ +&= \left\langle (\mathrm{pr}_X \circ f_W |_{W_{f_W}})^* \xi \cup \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}(\phi), (\pi_{f_W})_* \left( (\mathrm{pr}_{D(Q_q)} \circ f_W)^* \tau_q \cap t_{\overline{\nu_W}} \right) \right\rangle_{\overline{\nu_W}} \\ +&= \left\langle (\mathrm{pr}_X \circ f_W |_{W_{f_W}})^* \xi \cup \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}(\phi), t_{W_{f_W}, \tau_q} \right\rangle_{W_{f_W}} \\ +&= \left\langle \mathrm{cw}_{(g_W)_{f_W^* \nabla_d^p}}((\mathrm{pr}_X \circ f_W |_{W_{f_W}})^*\lambda), t_{W_{f_W}, \tau_q} \right\rangle_{W_{f_W}}, +\end{align*} +$$ + +where the first equality is (5.42), the second used (5.43), the fourth used +(3.21), the last used (5.44). Thus we get (5.41), and it follows that (5.39) +holds for nice bordism $(W, g_W, t_W, f_W)$ for nice $(M, g_M, t_M, f_M)$ with com- +patible tubular neighborhood structures. + +For general bordism $(W,g_W,t_W,f_W)$ which does not necessarily satisfy +the niceness condition, but is bounded by a nice $(M,g_M,t_M,f_M)$, we may +always find another $t'_W$ and $f'_W$ on $W$ such that $(W,g_W,t'_W,f'_W)$ is a nice +bordism data for $(M,g_M,t_M,f_M)$. Using the integrality condition (5.37) as +in the proof of Proposition 4.20, we get + +$$ +\langle c w_{g_W}(f_W^* \hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda)), t_W \rangle = \langle c w_{g_W}((f'_W)^* \hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda)), t'_W \rangle (\text{mod } \mathbb{Z}), +$$ + +so we get (5.39) for every nice $(M, g_M, t_M, f_M)$ and every bordism data +$(W, g_W, t_W, f_W)$ for it. + +Since the group $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\chi_p(q)))$ is generated by the +two kinds of elements (A) and (B) explained above, the compatibility con- +dition in Definition 4.56 (1) (c) determines the value of $\hat{\mathcal{P}}_{\nabla,\tau,s}(h)$ for all +elements in $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\chi_p(q)))$. The well-definedness and +the compatibility condition follow from the integrality (5.37) and the com- +patibility condition for the cases already checked above. +---PAGE_BREAK--- + +**Definition 5.45.** Let $n$ be a nonnegative integer and $p$ and $q$ be positive integers. Fixing additional choices $\{\nabla_d^p\}_{d}$, $\tau_q$ and $s$ in (i), (ii) and (iii) above, we define a natural transformation + +$$ (5.46) \quad \hat{\mathcal{P}}_{q,\nabla,\tau,s}^{n,p} : (\widehat{I\Omega_{\text{sing}}^G})^n(-) \to (\widehat{I\mathbb{Z}_{\text{sing}}})^{n+q}(-\times \text{Thom}_\infty(Q_q|\mathcal{X}^p(q))) $$ + +by, for each pair of topological spaces $(X, A)$ and $(\lambda, h) \in (\widehat{I\Omega_{\text{sing}}^G})^n(X, A)$, +the formula $\hat{\mathcal{P}}_{q,\nabla,\tau,s}^{n,p}(\lambda, h) := (\hat{\mathcal{P}}_{\nabla,\tau,s}(\lambda), \hat{\mathcal{P}}_{\nabla,\tau,s}(h))$, where the right hand side +is constructed above. We easily see that it is natural in $(X, A)$. + +**Lemma 5.47.** The transformation 5.46 induces a natural transformation from $(I\Omega_{\text{sing}}^G)^n(-)$ to $(I\mathbb{Z}_{\text{sing}})^{n+q}(- \times \text{Thom}_\infty(Q_q|\mathcal{X}^p(q)))$. Moreover, the resulting transformation does not depend on the choice (i), (ii) or (iii). + +*Proof.* To prove the first statement, it is enough to show that the following diagram commutes. + +$$ +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] +(5.48) \quad \begin{array}{c} (\widehat{I\Omega_{\text{sing}}^G})^n(-) \\ \xleftarrow[\text{a}_{\text{sing}}]{a} \xrightarrow[\text{a}_{\text{sing}}]{a} \end{array} \arrow[r] \quad & (\widehat{I\mathbb{Z}_{\text{sing}}})^{n+q}(- \times \text{Thom}_\infty(Q_q|\mathcal{X}^p(q))), \\ +& C^n(-; V_{I\Omega^G}^\bullet) \arrow[r, "(\hat{\mathcal{P}}_{\nabla,\tau,s})"] & C^{n+q}(- \times \text{Thom}_\infty(Q_q|\mathcal{X}^p(q)); \mathbb{R}) +\arrow[d, "C"] +\arrow[above, "a", "c"'] +\arrow[below, "b", "c"'] +\end{tikzcd} +$$ + +where the bottom arrow is the cochain map in (5.36). Let us take $\alpha \in C^n(X, A; V_{I\Omega_G}^\bullet)$. We need to show that + +$$ (5.49) \quad (\hat{\mathcal{P}}_{\nabla,\tau,s}(\delta\alpha), \hat{\mathcal{P}}_{\nabla,\tau,s}(\mathrm{cw}(\alpha))) = (\delta\hat{\mathcal{P}}_{\nabla,\tau,s}(\alpha), \mathrm{cw}(\hat{\mathcal{P}}_{\nabla,\tau,s}(\alpha))). $$ + +The cocycle part coincides because (5.36) is a cochain map. Thus, as in the argument preceding Definition 5.45, it is enough to check that the evaluations of the both sides of (5.49) coincide on every nice element in $C_{\Delta,n+q-1}^{\mathrm{fr}}((X,A) \times \mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d+q)))$ for each $d$. But this coincidence can be checked by the same computation for that proving the equality (5.41), and the details are left to the reader. Thus we get the first statement. The second statement can be checked in a straightforward way. $\square$ + +**Definition 5.50.** We define a natural transformation, + +$$ (5.51) \quad \mathcal{P}_q^{*,p}: (I\Omega_{\text{sing}}^G)^*(-) \to (I\mathbb{Z}_{\text{sing}})^{*+q}(- \times \text{Thom}_{\infty}(Q_q|\mathcal{X}^p(q))), $$ + +induced by (5.46) using Lemma 5.47. + +Now we show that the transformation (5.51) is a transformation of generalized cohomology theories. + +**Lemma 5.52.** The transformation (5.51) is a natural transformation of generalized cohomology theories. + +*Proof.* We need to check that the transformation (5.51) commutes with $\delta$ +in Definition 4.62. Let $(X, A)$ be a pair of topological spaces. We need to +---PAGE_BREAK--- + +show that the following diagram commutes. + +$$ +\begin{array}{ccc} + & \mathcal{P}_q^{n,p} & \\ +( I\Omega_{\text{sing}}^G )^n (A) & \longrightarrow & ( I\mathbb{Z}_{\text{sing}} )^{n+q} ( (X,A) \times \text{Thom}_\infty (Q_q |_{\mathfrak{x}^p(q)}) ) \\ +\Big|_\delta & & \Big|_\delta \\ +( I\Omega_{\text{sing}}^G )^{n+1} (X,A) & \xrightarrow{\mathcal{P}_q^{n+1,p}} & +( I\mathbb{Z}_{\text{sing}} )^{n+q+1} ( (X,A) \times \text{Thom}_\infty (Q_q |_{\mathfrak{x}^p(q)}) ) +\end{array} +$$ + +Take an element $(\lambda, h) \in (\widetilde{I\Omega_{\text{sing}}^G})^n(A)$. We have + +$$ +(5.54) \qquad +\begin{aligned} +\hat{\mathcal{P}}_{q,\nabla,\tau,s}^{n+1,p} \circ \hat{\delta}(\lambda, h) &= (\hat{\mathcal{P}}_{\nabla,\tau,s}(\delta\bar{\lambda}), \hat{\mathcal{P}}_{\nabla,\tau,s}(-h \circ \partial + \mathrm{cw}(\bar{\lambda}))) \\ +&= (\delta(\hat{\mathcal{P}}_{\nabla,\tau,s}(\bar{\lambda})), \hat{\mathcal{P}}_{\nabla,\tau,s}(-h \circ \partial + \mathrm{cw}(\bar{\lambda}))) +\end{aligned} +$$ + +$$ +(5.55) \qquad \hat{\delta} \circ \hat{\mathcal{P}}_{q, \nabla, \tau, s}^{n, p}(\lambda, h) = (\delta(\overline{\hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda)}), -\hat{\mathcal{P}}_{\nabla, \tau, s}(h) \circ \partial + \mathrm{cw}(\overline{\hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda)})). +$$ + +Here the second equality of (5.54) follows from the fact that the transformation (5.34) is a cochain map. We claim that the difference between (5.54) and (5.55) is given by + +$$ +a_{\text{sing}} (\hat{\mathcal{P}}_{\nabla, \tau, s}(\bar{\lambda}) - \overline{\hat{\mathcal{P}}_{\nabla, \tau, s}(\lambda)}), +$$ + +which implies the desired result. It is equivalent to the following equality of +homomorphisms $C_{\Delta,n+q}^{fr}((X,A) \times \operatorname{Thom}_{\infty}(Q_q|\chi_p(q))) \to \mathbb{R}/\mathbb{Z}$, + +$$ +(5.56) \qquad \hat{\mathcal{P}}_{\nabla, \tau, s}(-h \circ \partial + cw(\bar{\lambda})) = -\hat{\mathcal{P}}_{\nabla, \tau, s}(h) \circ \partial + cw(\bar{\lambda}) (\hat{\mathcal{P}}_{\nabla, \tau, s}(\bar{\lambda})). +$$ + +Since we know that both sides are compatible with $\delta (\hat{\mathcal{P}}_{\nabla,\tau,s}(\bar{\lambda}))$, as in the argument preceding Definition 5.45, it is enough to check that the evaluations of the both sides of (5.56) coincide on every nice element $[M,g,t,f] \in C_{\Delta,n+q}^{fr}((X,A) \times \text{Thom}_{\infty}(Q_q|\chi_d^p(d+q)))$ for each $d$. + +For such nice $(M, g, t, f)$ with a tubular neighborhood structure $\pi_f$, by +(5.38) we have + +$$ +\begin{align*} +\hat{\mathcal{P}}_{\nabla, \tau, s} (-h \circ \partial + \mathrm{cw}(\bar{\lambda})) ([M, g, t, f]) &= (-h \circ \partial + \mathrm{cw}(\bar{\lambda})) (\mathrm{PT}_{\nabla_d^p, \tau_q} [M, g, t, f]) \\ +&= -h (\mathrm{PT}_{\nabla_d^p, \tau_q} (\partial[M, g, t, f])) + \mathrm{cw}(\bar{\lambda}) (\mathrm{PT}_{\nabla_d^p, \tau_q} [M, g, t, f]). +\end{align*} +$$ + +Here the second equality used (3.26). On the other hand, again using (5.38) +we have + +$$ +-\hat{\mathcal{P}}_{\nabla,\tau,s}(h) \circ \partial([M,g,t,f]) = -h (\mathrm{PT}_{\nabla_d^p,\tau_q}(\partial[M,g,t,f])). +$$ + +Thus it is enough to show that + +$$ +\mathrm{cw}(\bar{\lambda}) (\mathrm{PT}_{\nabla_d^p, \tau_q}[M, g, t, f]) = \mathrm{cw}(\hat{\mathcal{P}}_{\nabla, \tau, s}(\bar{\lambda}) ([M, g, t, f]). +$$ + +But this follows from the same computation for that proving the equality +(5.41), and the details are left to the reader. Thus we get the commutativity +of (5.53) and the proof is complete. +$\square$ +---PAGE_BREAK--- + +Now we relate the natural transformation $\mathcal{P}_q^{*,p}$ with $\mathcal{P}_{q+1}^{*,p+1}$ and $\mathcal{P}_{q+1}^{*,p}$. + +**Lemma 5.57.** For each pair of positive integers $p$ and $q$, the following diagram commutes. + +$$ +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] +(5.58) \quad & (I\Omega_{\text{sing}}^G)^* \arrow[r, "$\mathcal{P}_q^{*,p}$"] \arrow[d, "$\mathcal{P}_q^{*,p+1}$] & (IZ_{\text{sing}})^{*+q}(- \times \operatorname{Thom}_\infty(Q_q|_{X^p(q)})) \\ +& (I\Omega_{\text{sing}}^G)^* \arrow[r, "$\mathcal{P}_q^{*,p+1}$"] \arrow[u, "$\mathcal{P}_q^{*,p}$"] & (IZ_{\text{sing}})^{*+q}(- \times \operatorname{Thom}_\infty(Q_q|_{X^{p+1}(q)})) +\arrow[ul, "$\Rightarrow$] \arrow[ur, "$\Rightarrow$] +\end{tikzcd} +$$ + +Here the right vertical arrow is induced by the inclusion $\mathcal{X}^p(q) \hookrightarrow \mathcal{X}^{p+1}(q)$. + +*Proof.* When we constructed the bottom arrow in (5.58), we choose a sequence ${\nabla_d^{p+1}}_d$ of connections on ${E^{p+1}G_d \to B^{p+1}G_d}_d$ as in (i). If we fix such a choice, by restricting it to ${E^p G_d \to B^p G_d}_d$, we get a sequence ${\nabla_d^p}_d$ which satisfies the requirement in (i) for $p$. Using this choice, the diagram (5.58) commutes on the level of the refinements (5.46). $\square$ + +**Lemma 5.59.** For each pair of positive integers $p$ and $q$, the following diagram commutes. + +$$ +\begin{tikzcd}[column sep=2.8em, row sep=2.8em] +(5.60) \quad & (I\Omega_{\text{sing}}^G)^* \arrow[r, "$\mathcal{P}_q^{*,p}$] & (IZ_{\text{sing}})^{*+q}(- \times \operatorname{Thom}_\infty(Q_q|_{X^p(q)})) \\ +& \arrow[u, "(-1)^{n+q}_{\text{susp}}", "id"'] & (IZ_{\text{sing}})^{*+q+1}(- \times \operatorname{Thom}_\infty(Q_q|_{X^p(q)} \oplus \mathbb{R})) \\ +& (I\Omega_{\text{sing}}^G)^* \arrow[r, "$\mathcal{P}_{q+1}^{*,p}$] & (IZ_{\text{sing}})^{*+q+1}(- \times \operatorname{Thom}_\infty(Q_{q+1}|_{X^p(q+1)})) +\arrow[u, "$(\mathrm{id}\times i)^*$], \arrow[ur, "$\Rightarrow$], \arrow[ul, "$\Rightarrow$] +\end{tikzcd} +$$ + +Here the right lower vertical arrow is induced by the inclusion $i: \mathcal{X}^p(q) \hookrightarrow \mathcal{X}^p(q+1)$ and the identification $Q_{q+1}|_{\mathcal{X}^p(q)} = Q_q|_{\mathcal{X}^p(q)} \oplus \mathbb{R}$. + +To prove the lemma, it is convenient to have an explicit description of the suspension homomorphism in $IZ_{\text{sing}}$ using the geometric Pontryagin-Thom construction. We get such a description in the following lemma. + +**Lemma 5.61** (The suspension isomorphism in $IZ_{\text{sing}}^*$). Let $(X, A)$ be a CW-pair and $n$ be a nonnegative integer. Let $\tau \in Z^1(D^1, S^0; \mathbb{R})$ be a singular cocycle representing the fundamental class in $H^1(D^1, S^0; \mathbb{R})$. Regard $D^1 = [-1, 1]$ as a disk bundle of the vector bundle $\mathbb{R} \to \{\text{pt}\}$, equipped with the trivial geometric stable fr-structure. + +(1) For each element $(\lambda, h) \in (\widetilde{IZ}_{\text{sing}})^n(X, A)$, there exists a unique homomorphism + +$$ +(5.62) \quad \mathrm{PT}_{\tau}(h) : C_{\Delta,n}^{\mathrm{fr}}((X,A) \times (D^1,S^0)) \to \mathbb{R}/\mathbb{Z} +$$ + +with the following properties. + +• The pair $(\mathrm{pr}_X^* \lambda \cup \mathrm{pr}_{D^1}^* \tau, \mathrm{PT}_\tau(h))$ is an element in $(\widetilde{IZ}_{\text{sing}})^{n+1}((X,A) \times (D^1, S^0))$. +---PAGE_BREAK--- + +* For a nice element $[M, g, t, f] \in C_{\Delta,n}^{\text{fr}}((X, A) \times (D^1, S^0))$ with a tubular neighborhood structure $\pi_f$, we have + +$$ (5.63) \qquad \mathrm{PT}_{\tau}(h)([M, g, t, f]) = h(\mathrm{PT}_{\tau}(M, g, t, f)). $$ + +The right hand side is well-defined by Lemma 3.24 (1) and Lemma 4.61. + +(2) We get a well-defined homomorphism + +$$ (5.64) \qquad \widehat{\mathrm{susp}}'_{\tau}: (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^n(X, A) \to (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}((X, A) \times (D^1, S^0)), $$ + +by mapping $(\lambda, h)$ to $(-1)^n (\mathrm{pr}_X^*\lambda \cup \mathrm{pr}_{D^1\tau}^*\tau, \mathrm{PT}_\tau(h))$. It induces a homomorphism + +$$ (5.65) \qquad \mathrm{susp}' : (\widetilde{\mathcal{I}\mathcal{Z}}_{\mathrm{sing}})^n(X, A) \to (\widetilde{\mathcal{I}\mathcal{Z}}_{\mathrm{sing}})^{n+1}((X, A) \times (D^1, S^0)), $$ + +which is independent of the choice of $\tau$. + +(3) The homomorphism 5.65 coincides with the suspension isomorphism for the generalized cohomology theory $\widetilde{\mathcal{I}\mathcal{Z}}_{\mathrm{sing}}^*$. + +By (3) of this lemma, we are going to denote $\widehat{\mathrm{susp}}_\tau := \mathrm{susp}'_\tau$ and $\mathrm{susp} = \mathrm{susp}'$ after the proof of this lemma. + +*Proof of Lemma 5.61.* The desired homomorphism $\mathrm{PT}_\tau(h)$ is essentially the same as the homomorphism $\hat{\mathrm{P}}_{\nabla,\tau,s}(h)$ for $G = \text{fr} = \{\text{1}\}$ in Definition 5.45, and the homomorphisms (5.64) and (5.65) corresponds to the homomorphisms (5.46) and (5.51), respectively. So (1) and (2) is proved in the same way as the well-definedness of Definition 5.45 and Definition 5.50 proved above. + +We prove (3). By the same argument as in the proof of Proposition 4.114, it is enough to show in the case $A = \emptyset$. The strategy is to use the $S^1$-integration map $\int$ for $(\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^*$ constructed in Definition 4.64. By Proposition 4.114, we know that it indeed realizes the $S^1$-integration map in (2.9). We use the following notations for the standard maps, + +$$ i: (D^1, S^0) \to (S^1, \{\text{pt}\}), $$ + +$$ j: S^1 \to (S^1, \{\text{pt}\}). $$ + +It is enough to show the commutativity of the following diagram. + +$$ (5.66) \qquad \begin{tikzcd} \mathcal{I}\mathcal{Z}_{\text{sing}}^{n+1}(X \times (S^1, \{\text{pt}\})) & \xrightarrow{\text{(id}\times\text{id})^*} \mathcal{I}\mathcal{Z}_{\text{sing}}^{n+1}(X \times (D^1, S^0)) \\[2ex] \mathcal{I}\mathcal{Z}_{\text{sing}}^{n+1}(X \times S^1) & \xleftarrow{\text{(id}\times\text{j})^*} \mathcal{I}\mathcal{Z}_{\text{sing}}^n(X). \arrow[\ suspension ' ] \\[2ex] & \xrightarrow{\int} \mathcal{I}\mathcal{Z}_{\text{sing}}^{n+1}((X \times D^1, S^0)) \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{id})^*} \mathcal{I}\mathcal{Z}_{\text{sing}}^{n+1}(X \times (S^1, \{\text{pt}\})) & \xrightarrow{\text{(id}\times\text{id})^*} \mathcal{I}\mathcal{Z}_{\text{sing}}^{n+1}(X \times (D^1, S^0)) \end{tikzcd} $$ + +Choose a cocycle $\tau_{S^1} \in Z^1(S^1, \{\text{pt}\}; \mathbb{R})$ representing the fundamental class. We denote $\tau := i^*\tau_{S^1} \in Z^1(D^1, S^0; \mathbb{R})$. Consider the following diagram. + +$$ (5.67) \qquad \begin{tikzcd} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times (S^1, \{\mathrm{pt}\})) & \xrightarrow{\text{(id}\times\text{id})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times (D^1, S^0)) \\[2ex] (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times S^1) & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{id})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times (S^1, \{\mathrm{pt}\})) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times (D^1, S^0)) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{id})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times (S^1, S^0)) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{id})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times (D^1, S^0)) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^1) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{id})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times S^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{id})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times S^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{id})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times S^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n}(X). \arrow[\ suspension ' ] \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}})^{n+1}(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X \times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X\times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X\times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n})(X). \\[2ex] & \xleftarrow{\text{(id}\times\text{j})^*} (\widetilde{\mathcal{I}\mathcal{Z}_{\mathrm{sing}}}^{n+1})(X\times D^0) & \xrightarrow{\int} (\widetilde{\mathcal{T}_g^n})(X)\end{xcode} + +$$ +---PAGE_BREAK--- + +Then it is enough to show that, for each element $x \in IZ_{\text{sing}}^{n+1}(X \times (S^1, \{\text{pt}\}))$, +there exists a representative in $(\widehat{IZ}_{\text{sing}})^{n+1}(X \times (S^1, \{\text{pt}\}))$ which maps to +the same element in $(\widehat{IZ}_{\text{sing}})^{n+1}(X \times (D^1, S^0))$ under the two compositions +in (5.67). + +Take any element $x \in IZ_{\text{sing}}^{n+1}(X \times (S^1, \{\text{pt}\}))$. We can find an element +$\mathcal{X} \in (\widehat{IZ}_{\text{sing}})^{n+1}(X \times (S^1, \{\text{pt}\}))$ of the form + +$$ +\mathcal{X} = (-1)^n (\mathrm{pr}_X^* \lambda \cup \mathrm{pr}_{S^1}^* \tau_{S^1}, h_{X \times (S^1, \{\mathrm{pt}\})}) +$$ + +for some $\lambda \in Z^n(X)$, that satisfies $x = I_{\text{sing}}(\mathcal{X})$. This follows from the +suspension isomorphism in the ordinary cohomology. Now our goal is to +show + +$$ +(5.68) \qquad \widehat{\mathrm{susp}}'_{\tau} \left( \int (\mathrm{id} \times j)^* \mathcal{X} \right) = (\mathrm{id} \times i)^* \mathcal{X}. +$$ + +First we look at the left hand side of (5.68). We have + +$$ +\int (\mathrm{id} \times j)^* \chi = (\lambda, h_X), +$$ + +where we set $h_X := (-1)^n \int (\mathrm{id} \times j)^* h_{X \times (S^1, \{\mathrm{pt}\})}$ in the notation of Definition 4.64. Using the notations in Definition 4.64, we have + +$$ +(5.69) \qquad h_X([M, g, t, f]) = (-1)^{n-1} h_{X \times (S^1, \{\mathrm{pt}\})} (S^1 \times M, g_{S^1}, P_{S^1} t, \sigma \circ (\mathrm{id}_{S^1} \times f)). +$$ + +Here we denote by + +$$ +(5.70) \qquad \sigma : S^1 \times X \stackrel{\sim}{=} X \times S^1 +$$ + +the homeomorphism flipping the two factors. Applying $\widehat{\mathrm{susp}}'_{\tau}$ we get + +$$ +(5.71) \qquad \widehat{\mathrm{susp}}'_{\tau} \left( \int \mathcal{X} \right) = (-1)^n (\mathrm{pr}_X^* \lambda \cup \mathrm{pr}_{D^1}^* \tau, \mathrm{PT}_{\tau}(h_X)). +$$ + +On the other hand the right hand side of (5.68) is just + +$$ +(5.72) \qquad (\mathrm{id} \times i)^* \chi = (-1)^n (\mathrm{pr}_X^* \lambda \cup \mathrm{pr}_{D^1}^* \tau, (\mathrm{id} \times i)^* h_{X \times (S^1, \{\mathrm{pt}\})}). +$$ + +Since the cocycle parts of (5.71) and (5.72) coincide, we only need to check +that the evaluation of $\mathrm{PT}_{\tau}(h_X)$ and $(\mathrm{id} \times i)^*\tilde{h}_{X\times(S^1,\{\mathrm{pt}\})}$ coincide on a rep- +resentative in $C_{\Delta,n}^{\mathrm{fr}}(X\times(D^1,S^0))$ for each element in the stably framed +bordism group $\Omega_n^{\mathrm{fr}}(X\times(D^1,S^0))$. By the suspension isomorphism $\Omega_n^{\mathrm{fr}}(X\times$ +$(D^1,S^0)) \simeq \Omega_{n-1}^{\mathrm{fr}}(X)$, we can take it as + +$$ +(5.73) \qquad [D^1 \times M, g_{D^1}, P_{D^1} t, \sigma \circ (\chi \times f)] \in C_{\Delta,n}^{\text{fr}}(X \times (D^1, S^0)), +$$ + +for some element $[M,g,t,f] \in C_{\Delta,n-1}^{fr}(X)$. Here $\sigma: (D^1, S^0) \times X \simeq X \times (D^1, S^0)$ is the flip (we use the same notation as (5.70)), and we take a smooth map $\chi: (D^1, S^0) \to (D^1, S^0)$ such that it is homotopic to $\text{id}_{D^1}$, $\chi^{-1}(\{-1\}) = [-1, -0.5]$ and $\chi^{-1}(\{1\}) = [0.5, 1]$.¹³ The desired result follows from the following two claims. + +¹³The reason for introducing $\chi$ is to make the map compatible with the collar structure. +---PAGE_BREAK--- + +**Claim 5.74.** We have + +$$ \mathrm{PT}_{\tau}(h_X)([D^1 \times M, g_{D^1}, P_{D^1}t, \sigma \circ (\chi \times f)]) = (-1)^{n-1}h_X([M, g, t, f]). $$ + +*Proof of Claim 5.74.* We need to be careful because the element (5.73) is not nice. Namely the fundamental chain $P_{D^1}t$ does not satisfy Condition (C) in Definition 3.15. Instead, $(D^1 \times M, g_{D^1}, P_{[-1,-0.5]}t + P_{[-0.5,0.5]}t + P_{[0.5,1]}t, \chi \times f)$ is nice with the tubular neighborhood structure $(-0.5, 0.5) \times M \to M$. We have + +(5.75) + +$$ \begin{aligned} & \mathrm{PT}_{\tau}(h_X)([D^1 \times M, g_{D^1}, P_{D^1}t, \sigma \circ (\chi \times f)]) \\ &= \mathrm{PT}_{\tau}(h_X)([D^1 \times M, g_{D^1}, P_{[-1,-0.5]}t + P_{[-0.5,0.5]}t + P_{[0.5,1]}t, \sigma \circ (\chi \times f)]). \end{aligned} $$ + +This is checked by taking a bordism data, where only the nontrivial part is the fundamental chain, on which the evaluation of the pullback of $\mathrm{pr}_X^*\lambda \cup \mathrm{pr}_{D^1}^\tau$ vanishes. For this check it is useful to notice that $f^*\lambda \in Z^n(M; \mathbb{R})$ is a coboundary, $f^*\lambda = \delta\alpha$ for some $\alpha \in C^{n-1}(M; \mathbb{R})$, by the dimensional reason that $\dim M = n - 1$. + +By (5.63) we have + +(5.76) + +$$ \begin{align*} & \mathrm{PT}_{\tau}(h_X)([D^1 \times M, g_{D^1}, P_{[-1,-0.5]}t + P_{[-0.5,0.5]}t + P_{[0.5,1]}t, \sigma \circ (\chi \times f)]) \\ &= h_X(\mathrm{PT}_{\tau}(D^1 \times M, g_{D^1}, P_{[-1,-0.5]}t + P_{[-0.5,0.5]}t + P_{[0.5,1]}t, \sigma \circ (\chi \times f))) \\ &= (-1)^{n-1}h_X([M, g, t, f]). \end{align*} $$ + +The sign $(-1)^{n-1}$ follows by Lemma 3.24 (2) applied to $q=0$ and Lemma 4.61. Indeed, the stable framing $\psi^{\mathrm{fr}} \oplus \mathrm{id}_{\mathbb{R}}: \mathbb{R}^k \oplus TM \oplus \mathbb{R} \to \mathbb{R}^{k+n+q+1}$ appearing in the statement of Lemma 3.24 (2) differs from $g_{D^1}$ by exchanging the order of $TM$ and $\mathbb{R}$. By (5.75) and (5.76) we get Claim 5.74. $\square$ + +**Claim 5.77.** We have + +$$ (\mathrm{id} \times i)^* h_{X \times (S^1, \{\mathrm{pt}\})} ([D^1 \times M, g_{D^1}, P_{D^1}t, \sigma \circ (\chi \times f)]) = (-1)^{n-1} h_X([M, g, t, f]). $$ + +*Proof of Claim 5.77.* Denote by $\chi_{S^1}: S^1 \to S^1$ the map induced from $\chi$. We have + +$$ \begin{align*} & (\mathrm{id} \times i)^* h_{X \times (S^1, \{\mathrm{pt}\})} ([D^1 \times M, g_{D^1}, P_{D^1}t, \sigma \circ (\chi \times f)]) \\ &= h_{X \times (S^1, \{\mathrm{pt}\})} ([D^1 \times M, g_{D^1}, P_{D^1}t, \sigma \circ ((i \circ \chi) \times f)]) \\ &= h_{X \times (S^1, \{\mathrm{pt}\})} ([S^1 \times M, g_{S^1}, P_{S^1}t, \sigma \circ (\chi_{S^1} \times f)]). \end{align*} $$ + +Here the last equality used the gluing formula in Proposition 4.92, Remark 4.34 and Lemma 4.61. By (5.69), it is enough to show that + +$$ h_{X \times S^1} ([S^1 \times M, g_{S^1}, P_{S^1}t, \sigma \circ (\chi_{S^1} \times f)]) = h_{X \times S^1} ([S^1 \times M, g_{S^1}, P_{S^1}t, \sigma \circ (\mathrm{id} \times f)]). $$ + +But it follows by considering the bordism data $(D^1 \times S^1 \times M, (g_{S^1})_{D^1}, P_{D^1}(P_{S^1}t), \sigma \circ (\widetilde{\chi}_{S^1} \times f))$, where $\widetilde{\chi}_{S^1}$ is a homotopy from $\chi_{S^1}$ to $\mathrm{id}_{S^1}$ and applying the compatibility condition. The evaluation of $\mathrm{pr}_X^*\lambda \cup \mathrm{pr}_{S^1}^\tau\tau_{S^1}$ on this bordism data vanishes. This completes the proof of Claim 5.77. $\square$ + +By Claim 5.74 and Claim 5.77 we get (5.68) and this completes the proof of Lemma 5.61. $\square$ +---PAGE_BREAK--- + +*Proof of Lemma 5.59.* Fix a CW-pair $(X, A)$ and an element $(\lambda, h) \in (\widetilde{I\Omega_{\text{sing}}^{G}})^n(X, A)$. +Also fix a homeomorphism + +$$ \xi: \operatorname{Thom}_{\infty}(Q_q|\mathcal{X}^p(q) \oplus \mathbb{R}) \simeq \operatorname{Thom}_{\infty}(Q_q|\mathcal{X}^p(q)) \times (D^1, S^0) $$ + +which preserves the fiber and smooth on the interior. Choosing data (i), (ii), (iii) and $\tau$ in Lemma 5.61, we need to compare its images in $(\widetilde{IZ}_{\text{sing}})^{n+q+1}((X, A) \times \operatorname{Thom}_{\infty}(Q_q|\mathcal{X}^p(q) \oplus \mathbb{R}))$ under two compositions in (5.60). For (ii), we may choose a Thom cocycle $\tau_{q+1} \in Z^{q+1}(\operatorname{Thom}_{\infty}(Q_{q+1})|\mathcal{X}^p(q+1))$; $\operatorname{Ori}(Q_{q+1})$ so that its restriction to $Q_{q+1}|\mathcal{X}^p(q) = Q_q|\mathcal{X}^p(q) \oplus \mathbb{R}$ is of the form $\xi^*(\tau_q \cup \tau_{\mathbb{R}})$, where $\tau_q \in Z^q(\operatorname{Thom}_{\infty}(Q_q|\mathcal{X}^p(q))$; $\operatorname{Ori}(Q_q)$) and $\tau_{\mathbb{R}} \in Z^q(D^1, S^0; \mathbb{R})$ are Thom cocycles for $Q_q|\mathcal{X}^p(q)$ and $\mathbb{R}$, respectively. For (iii), a choice $s$ for $q+1$ restricts to a choice for $q$. Using such choices, we claim that + +$$ (-1)^{n+q} (\mathrm{id} \times \xi)^* \widehat{\mathrm{susp}}_{\tau_{\mathbb{R}}} \circ \hat{\mathcal{P}}_{q,\nabla,\tau_q,s}^{n,p} (\lambda, h) = (\mathrm{id} \times i)^* \circ \hat{\mathcal{P}}_{q+1,\nabla,\tau_{q+1},s}^{n,p} (\lambda, h). $$ + +The coincidence of the cocycle part easily follows from our choices for $\tau_{q+1}$ and $s$. Once we know it, we only need to show that the evaluation of the both sides of (5.78) on nice elements in $C_{\Delta,n+q}^{\mathrm{fr}}((X,A)\times\mathrm{Thom}_{\infty}(Q_q|\mathcal{X}_d^p(d+q)))$ for each $d$ agrees. To show it, we need to compare the following geometric Pontryagin-Thom constructions, + +$$ \mathrm{PT}_{\nabla_d^p, \tau_q} \circ \mathrm{PT}_{\tau_{\mathbb{R}}} \circ (\mathrm{id} \times \xi)_* \quad \text{and} \quad \mathrm{PT}_{\nabla_d^p, \tau_{q+1}}. $$ + +They produce different elements in $C_{\Delta,n-1}^G(X, A)$, but by Lemma 3.24 (2), the difference is as in Lemma 4.61. So the value of $h$ are the same by Lemma 4.61. This completes the proof. $\square$ + +For each nonnegative integer $n$, when $p \ge n+1$ and $q \ge n+1, 2$ we have +$(\mathrm{id}_- \wedge w_q^p)^* \circ \mathrm{susp}^q : (\mathrm{IZ}_{\mathrm{sing}})^n (-\wedge \mathrm{MTG}) \simeq (\mathrm{IZ}_{\mathrm{sing}})^{n+q} (-\wedge \mathrm{Thom}(Q_q|\mathcal{X}^p(q)), \{\mathrm{pt}\})$ +on $\mathrm{CW}_*^{\mathrm{op}}$ by (5.30) and Theorem 5.1. Composing it with $\mathcal{P}_q^{n,p}$, we get a +natural transformation + +$$ (-1)^{nq + \frac{q(q-1)}{2}} ((\mathrm{id}_- \wedge w_q^p)^* \circ \mathrm{susp}^q)^{-1} \circ \mathcal{P}_q^{n,p} : (\widetilde{I\Omega}_{\mathrm{sing}}^G)^n \to (\widetilde{I\Omega}_{\mathrm{sing}})^n (-\wedge \mathrm{MTG}). $$ + +Moreover, by Lemma 5.57 and Lemma 5.59, the transformation (5.79) does not depend on $p$ and $q$ satisfying $p \ge n+1$ and $q \ge n+1, 2$. Note that the sign in (5.79) is $nq + \frac{q(q-1)}{2} = n + (n+1) + \cdots + (n+q-1)$ which comes from the sign in (5.60). Thus we define the following. + +**Definition 5.80.** We define a natural transformation of generalized cohomology theories, + +$$ \mathcal{P}: (\widetilde{I\Omega}_{\text{sing}}^{G})^{*} \rightarrow (\widetilde{I\mathbb{Z}}_{\text{sing}})^{*}(-\wedge \mathrm{MTG}) $$ + +by (5.79) for any choice of $p \ge *+1$ and $q \ge *+1, 2$ for each degree $*$ $\ge 0$, and zero for negative degrees. It is indeed a transformation of generalized cohomology theories since each $\mathcal{P}_q^{*,p}$ is (Lemma 5.52). + +Finally we are going to show that Definition (5.80) gives the desired isomorphism. For this, we use the following lemma. +---PAGE_BREAK--- + +**Lemma 5.81.** For each integer $n$, the following diagram commutes. + +(5.82) + +$$ +\begin{tikzcd} +& & \operatorname{Hom}(\Omega_{n-1}^G(-), \mathbb{R}/\mathbb{Z}) \arrow[r, "$p_{\mathrm{sing}}$] & (\mathrm{IZ}_{\mathrm{sing}})^n(-) \arrow[r, "$\mathrm{ch}_{\mathrm{sing}}$] & \operatorname{Hom}(\Omega_n^G(-), \mathbb{R}) \\ +& & \colvert & \colvert & \colvert \\ +& & \operatorname{Hom}(\Omega_{n-1}^{\mathrm{fr}}(- \wedge \mathrm{MTG}), \mathbb{R}/\mathbb{Z}) \arrow[r, "$p_{\mathrm{sing}}$] & (\mathrm{IZ}_{\mathrm{sing}})^n(- \wedge \mathrm{MTG}) \arrow[r, "$\mathrm{ch}_{\mathrm{sing}}$] & \operatorname{Hom}(\Omega_n^{\mathrm{fr}}(- \wedge \mathrm{MTG}), \mathbb{R}) +\end{tikzcd} +$$ + +Here the right and the left vertical arrows are induced by the isomorphism $\Omega_{\bullet}^{G}(-) \simeq \Omega_{\bullet}^{\text{fr}}(- \wedge \text{MTG})$ by the Pontryagin-Thom construction. + +*Proof.* This is easy to see by the construction of $\mathcal{P}$. $\square$ + +*Proof of Theorem 5.21.* By Lemma 5.81 and Proposition 4.67, the five lemma implies that $\mathcal{P}$ gives the isomorphism of generalized cohomology theories, + +$$ +\mathcal{P}: (I\Omega_{\text{sing}}^G)^* \simeq (\text{IZ}_{\text{sing}})^*(-\wedge \text{MTG}) +$$ + +Since we have $(\mathcal{I}\mathcal{Z}_{\text{sing}})^*(-\wedge \mathcal{M}\mathcal{T}\mathcal{G}) \simeq \mathcal{I}\mathcal{Z}^*(-\wedge \mathcal{M}\mathcal{T}\mathcal{G})$ by Theorem 5.1 and $\mathcal{I}\mathcal{Z}^*(-\wedge \mathcal{M}\mathcal{T}\mathcal{G}) \simeq (\mathcal{I}\Omega^G)^*$ by definition, composing them we get the desired isomorphism $(\mathcal{I}\Omega_{\text{sing}}^G)^* \simeq (\mathcal{I}\Omega^G)^*$. This finishes the proof of Theorem 5.21. $\square$ + +5.2.3. The differential version of the transformation $(I\Omega^G)^* \to I\mathcal{Z}^*(-\wedge \mathcal{M}\mathcal{T}\mathcal{G})$. In this subsection, we explain the differential version of the transformations given in Definition 5.50, in a simplified setting. The construction is much simpler in the differential case, so the authors hope that it helps the reader to understand Subsection 5.2. We also use this to analyze the examples in Section 6. + +We start from the setting in Subsection 3.2, where we are given a principal $G_d$-bundle with connection $(P, \nabla)$ on a manifold $B$, with a rank-$q$ vector bundle $\pi: W_q \to B$ with fixed isomorphism of vector bundles, + +$$ +(5.83) \qquad (P \times_{\rho_d} \mathbb{R}^d) \oplus W_q \simeq \mathbb{R}^{d+q}. +$$ + +We also fix a Thom form $\tau \in \Omega_{\mathrm{clo}}^q(\mathrm{Thom}_{\infty}(W_q); \mathrm{Ori}(\pi))$. + +**Definition 5.84.** We define a natural transformation of functors $Mfd^{op} \to \mathrm{Ab}^\mathbb{Z}$, + +$$ +(5.85) \qquad \hat{\mathcal{P}}_{\mathrm{dR},\nabla,\tau} : (\widehat{\Omega_{\mathrm{dR}}^{G_d}})^* \to (\widehat{\mathrm{IZ}_{\mathrm{dR}}})^{*+q} (- \times \mathrm{Thom}_\infty(W_q)) +$$ + +as follows. Fix a manifold $X$ and a nonnegative integer $n$, and take an element $(\omega, h) \in (\widehat{\Omega_{\mathrm{dR}}^{G_d}})^n(X)$. We set + +$$ +(5.86) \quad \hat{\mathcal{P}}_{\mathrm{dR},\nabla,\tau}(\omega,h) := (\pi^*\mathrm{cw}\nabla(\mathrm{pr}_X^*\omega) \wedge \mathrm{pr}_{D(W_q)}^\tau, \mathrm{PT}_{\mathrm{dR},\nabla,\tau}(h)), +$$ + +where + +• The map $\mathrm{pr}_X: X \times B \to X$ pulls back $\omega$ to $\mathrm{pr}_X\omega \in \Omega_{\mathrm{clo}}^n(X \times B; V_{I\Omega^{G_d}})$. Applying the homomorphism (4.7), we get $\mathrm{cw}\nabla(\mathrm{pr}_X^*\omega) \in \Omega_{\mathrm{clo}}^n(X \times B; \mathrm{Ori}(\pi))$, where we have used the identification $P \times_{G_d} \mathbb{R}_{G_d} \simeq \mathrm{Ori}(\pi)$ and denoted the pullback of $\nabla$ to $X \times B$ also by $\nabla$. This gives $\pi^*\mathrm{cw}\nabla(\mathrm{pr}_X^*\omega) \wedge \mathrm{pr}_{D(W_q)}^\tau \in \Omega_{\mathrm{clo}}^{n+q}(X \times \mathrm{Thom}_\infty(W_q))$. +---PAGE_BREAK--- + +• The homomorphism $\mathrm{PT}_{\mathrm{dR},\nabla,\tau}(h) : \mathcal{C}^{\mathrm{fr}}_{\infty,n+q-1}(X \times \mathrm{Thom}_{\infty}(\mathcal{W}_q)) \to \mathbb{R}/\mathbb{Z}$ is characterized by the following two properties. + +– If $(M, g, f) \in \mathcal{C}^{\mathrm{fr}}_{\infty,n+q-1}(X \times \mathrm{Thom}_{\infty}(\mathcal{W}_q))$ is nice with a tubular neighborhood structure $\pi_f$, we have + +$$ (5.87) \qquad \mathrm{PT}_{\mathrm{dR},\nabla,\tau}(h)([M,g,f]) = h(\mathrm{PT}_{\nabla}[M,g,f]). $$ + +Here $\mathrm{PT}_{\nabla}[M,g,f] \in \mathcal{C}^{G_d}_{\infty,n-1}(X)$ is obtained by the geometric Pontryagin-Thom construction (Definition 3.22). + +– It is compatible with $\pi^*\mathrm{cw}\nabla(\mathrm{pr}_X^*\omega) \wedge \mathrm{pr}_{D(W_q)}^\tau$ in the sense of Definition 4.11 (1) (c). + +The existence and the uniqueness of the homomorphism $\mathrm{PT}_{\mathrm{dR},\nabla,\tau}(h)$ satisfying the above properties can be shown in the same way as we did for Definition 5.45. It is easily seen that the homomorphism $\hat{\mathcal{P}}_{\mathrm{dR},\nabla,\tau}$ is natural in X. + +Analogously to Lemma 5.47, the natural transformation (5.85) reduces to the natural transformation on the level of cohomology theories, and it is independent of the choices of $\nabla$ and $\tau$. This allows us to define the following. + +**Definition 5.88.** Given a principal $G_d$-bundle $P$ on a manifold $B$ and a vector bundle $\pi: W_q \to B$ with fixed isomorphism (5.83), We define the natural transformation + +$$ (5.89) \qquad \mathcal{P}_{\mathrm{dR},P}: (I\Omega_{\mathrm{dR}}^{G_d})^* \to (I\mathbb{Z}_{\mathrm{dR}})^{*+q}(-\times \mathrm{Thom}_{\infty}(\mathcal{W}_q)) $$ + +to be the transformation induced from (5.85) by any choice of $\nabla$ and $\tau$. + +Now we relate the construction here with that of Subsubsection 5.2.2. Let $G_d$ be the $d$-component of a sequence $G$, and assume we are given $P$, $B$ and $W_q$ as above. We get the classifying element, + +$$ (5.90) \qquad f \in [\Sigma^\infty\mathrm{Thom}(\mathcal{W}_q), \Sigma^q MTG]. $$ + +Namely, the map $f$ comes from a pullback diagram of vector bundles + +$$ (5.91) \qquad \begin{tikzcd} W_q \arrow[r, "f"] & Q_q \\ \arrow[d, "π"] & \\ B \arrow[r, "-"'] & X_d(d+q) \end{tikzcd} $$ + +for which the bottom horizontal arrow is a classifying map for $P$, which pulls back the isomorphism (5.83). + +We denote the composition of the forgetful homomorphism $(I\Omega_{\mathrm{dR}}^G)^* \to (I\Omega_{\mathrm{dR}}^{G_d})^*$ with (5.89) also by $\mathcal{P}_{\mathrm{dR},P}$. We have the following. + +**Proposition 5.92.** In the above setting, the following diagram of natural transformations commutes. + +$$ (5.93) \qquad \begin{tikzcd} (I\Omega_{\mathrm{sing}}^G)^n \arrow[r, "P", "simeq"] & (I\mathbb{Z}_{\mathrm{sing}})^n (-\wedge MTG) \\ \arrow[d, "P"] & (-1)^{nq+\frac{q(q-1)}{2}} f^*\circ\mathrm{susp}^q \\ (I\Omega_{\mathrm{dR}}^G)^n \arrow[r, "P_{\mathrm{dR},P}"] & (I\mathbb{Z}_{\mathrm{dR}})^{n+q} (-\times \mathrm{Thom}_{\infty}(\mathcal{W}_q)) \end{tikzcd}. $$ +---PAGE_BREAK--- + +*Proof.* By the construction of the classifying map (5.90), it is enough to show in the case where $B = \mathcal{X}_d^p(d+q)$ for some $p$ with the principal $G_d$-bundle and $W_q = Q_q$ as in the last subsections. We also fix a $G_d$-connection $\nabla$ and a Thom form $\tau \in \Omega_{\mathrm{clo}}^q(\mathrm{Thom}_{\infty}(W_q); \mathrm{Ori}(\pi))$. + +Recall that, in Proposition 4.70 and its proof, the isomorphism $(I\Omega_{\mathrm{sing}}^G)^* \simeq (I\Omega_{\mathrm{dR}}^G)^*$ on manifolds is given by passing to the smooth singular model and constructing homomorphisms $(I\widehat{\Omega}_{\mathrm{dR}}^G)^* \to (I\widehat{\Omega}_{\mathrm{sing},\infty}^G)^*$ and $(I\widehat{\Omega}_{\mathrm{sing}}^G)^* \to (I\widehat{\Omega}_{\mathrm{sing},\infty}^G)^*$ which induces the isomorphisms on the cohomology level. The latter is just the forgetful homomorphism, but the former used a choice of natural cochain homotopy $B$ connecting $\wedge$ on differential forms and $\cup$ on smooth singular cochains (2.44). So from now on we fix such a natural cochain homotopy $B$. As explained in Remark 4.55, the Chern-Weil construction in the differential models and the (smooth) singular models are different. To distinguish them we denote the former by $cw^{\mathrm{dR}}$ and the latter by $cw^{\mathrm{sing}}$. + +Passing from the singular model to the smooth singular model is also easy here. Recall that the transformation $\mathcal{P}$ in the top of (5.93) is constructed by transformation $\mathcal{P}_q^{n,p}$ in Definition 5.50. Recalling its construction, its pullback to $\mathrm{Thom}_{\infty}(W_q)$ can also be constructed in the smooth singular version in the way that it is compatible with the singular version under the forgetful map. Namely, it is a transformation + +$$ (5.94) \quad \hat{\mathcal{P}}_{\mathrm{sing},\infty,\nabla,\tau}: (I\widehat{\Omega}_{\mathrm{sing},\infty}^G)^n \to (I\widehat{\mathbb{Z}}_{\mathrm{sing},\infty})^{n+q} (-\times \mathrm{Thom}_\infty(W_q)), $$ + +which is defined by, given $(\lambda, h) \in (\widehat{I\Omega}_{\mathrm{sing},\infty}^G)^n(X)$, + +$$ (5.95) \quad \hat{\mathcal{P}}_{\mathrm{sing},\infty,\nabla,\tau}(\lambda,h) := (\pi^*\mathrm{cw}_{\nabla}^{\mathrm{sing}}(\mathrm{pr}_X^*\lambda) \cup \mathrm{pr}_D^*(W_q)\tau, \mathrm{PT}_{\mathrm{sing},\infty,\nabla,\tau}(h)), $$ + +where $\mathrm{PT}_{\mathrm{sing},\infty,\nabla,\tau}(h)$ is the unique homomorphism characterized by the property that + +$$ (5.96) \quad \mathrm{PT}_{\mathrm{sing},\infty,\nabla,\tau}(h)([M,g,t,f]) = h(\mathrm{PT}_{\nabla,\tau}[M,g,t,f]) $$ + +if $[M,g,t,f]$ is nice, and it is compatible with $\pi^*\mathrm{cw}_{\nabla}^{\mathrm{sing}}(\mathrm{pr}_X^*\omega) \cup \mathrm{pr}_D^*(W_q)\tau$. + +To ease the notation, in this proof we drop the reference to $P$, $\nabla$, $\tau$ where there is no fear of confusion. We denote objects in differential models by the symbol “dR” and smooth singular models by “sing, $\infty$”. + +Now we have the following diagram. + +$$ (5.97) \quad \begin{tikzcd} (I\widehat{\Omega}_{\mathrm{sing},\infty}^G)^n \arrow[r, " \hat{\mathcal{P}}_{\mathrm{sing},\infty}"] & (I\widehat{\mathbb{Z}}_{\mathrm{sing},\infty})^n (- \times \mathrm{Thom}_\infty(W_q)) \\[1em] (I\widehat{\Omega}_{\mathrm{dR}}^G)^n \arrow[u,r, " \hat{\mathcal{P}}_{\mathrm{dR}}"'] & (I\widehat{\mathbb{Z}}_{\mathrm{dR}})^{n+q} (- \times \mathrm{Thom}_\infty(W_q)) \arrow[u,r, " "'] \end{tikzcd} $$ + +We are reduced to show that (5.97) commutes on the cohomology level. + +Fix an element $(\omega, h_{\mathrm{dR}}) \in (\widehat{I\Omega}_{\mathrm{dR}}^G)^n(X)$. As explained in the proof of Proposition 4.70, the left vertical arrow of (5.97) sends it to $(\omega, h_{\mathrm{sing},\infty})$, +---PAGE_BREAK--- + +where $h_{\text{sing},\infty}$ is given by (see (4.78)) + +$$ (5.98) \qquad h_{\text{sing},\infty}([M, g, t, f]) = h_{\text{dR}}([M, g, f]) - \langle B_g(f^*\omega), t \rangle_M, $$ + +where $B_g$ is constructed from the cochain homotopy $B$ by the formula (4.76). Now apply the top vertical arrow in (5.97) to it. Then we get + +$$ (5.99) \qquad \hat{P}_{\text{sing},\infty}(\omega, h_{\text{sing},\infty}) = (\pi^* \text{cw}_{\nabla}^{\text{sing}}(\text{pr}_X^*\omega) \cup \text{pr}_{D(W_q)}^*\tau, \text{PT}_{\text{sing},\infty}(h_{\text{sing},\infty})). $$ + +On the other hand, applying the right vertical arrow to the element (5.86), we get + +$$ (5.100) \qquad (\pi^* \text{cw}_{\nabla}^{\text{dR}}(\text{pr}_X^*\omega) \wedge \text{pr}_{D(W_q)}^*\tau, \text{PT}_{\text{dR}}(h_{\text{dR}}) \circ \text{fgt}), $$ + +where $\text{fgt}: C_{\Delta,\infty,n+q-1}^{\text{fr}}(X \times \text{Thom}_{\infty}(W_q)) \to C_{\infty,n+q-1}^{\text{fr}}(X \times \text{Thom}_{\infty}(W_q))$ is the forgetful map given by sending $[M,g,t,f]$ to $[M,g,f]$. The modification by $B$ as in (5.98) does not appear in $\mathbb{Z}$, as explained in the proof of Proposition 4.137. + +We have to show that the two elements (5.99) and (5.100) differ by an image of $a_{\text{sing},\infty}$. Indeed, we claim that we have + +$$ (5.101) \qquad \left( \pi^* \text{cw}_{\nabla}^{\text{dR}}(\text{pr}_X^*\omega) \wedge \text{pr}_{D(W_q)}^*\tau, \text{PT}_{\text{dR}}(h_{\text{dR}}) \circ \text{fgt} \right) - \left( \pi^* \text{cw}_{\nabla}^{\text{sing}}(\text{pr}_X^*\omega) \cup \text{pr}_{D(W_q)}^*\tau, \text{PT}_{\text{sing},\infty}(h_{\text{sing},\infty}) \right) \\ = a_{\text{sing},\infty} \left( B(\pi^* \text{cw}_{\nabla}^{\text{dR}}(\text{pr}_X^*\omega), \text{pr}_{D(W_q)}^*\tau) + \pi^* B_{\nabla}(\text{pr}_X^*\omega) \cup \text{pr}_{D(W_q)}^*\tau \right), $$ + +where $B_{\nabla}$ is defined in (4.73). By (2.45) and (4.75), we see that + +$$ (5.102) \qquad \begin{aligned} & \pi^* \text{cw}_{\nabla}^{\text{dR}}(\text{pr}_X^*\omega) \wedge \text{pr}_{D(W_q)}^*\tau - \pi^* \text{cw}_{\nabla}^{\text{sing}}(\text{pr}_X^*\omega) \cup \text{pr}_{D(W_q)}^*\tau \\ = & \delta \left( B(\pi^* \text{cw}_{\nabla}^{\text{dR}}(\text{pr}_X^*\omega), \tau) + \pi^* B_{\nabla}(\text{pr}_X^*\omega) \cup \tau \right), \end{aligned} $$ + +so the cocycle parts of (5.101) coincide. Once we know it, we are left to check that the evaluation of both sides of (5.101) on nice elements in $C_{\Delta,\infty,n+q-1}^{\text{fr}}(X \times \text{Thom}_{\infty}(W_q))$ coincide. Let $(M,g,t,f)$ be nice with a tubular neighborhood structure $\pi_f: \nu \to M_f$. We use the notations in Subsection 3.2 and denote $\text{PT}_{\nabla,\tau}(M,g,t,f) := (M_f, g_{f*\nabla}, t_{M_f,\tau}, \text{pr}_X \circ (f|_{M_f}))$. In particular we denote by $t_{M_f,\tau}$ the fundamental chain on $M_f$ in (3.21), given by + +$$ (5.103) \qquad t_{M_f, \tau} := (\pi_f)_* ((\text{pr}_{D(W_q)} \circ f)^* \tau \cap t_{\bar{\nu}}) \in C_{\infty, n-1}(M_f; \operatorname{Ori}(M_f)) $$ + +Then by (5.87) we have + +$$ \mathrm{PT}_{\mathrm{dR}}(h_{\mathrm{dR}}) \circ \mathrm{fgt}([M, g, t, f]) = h_{\mathrm{dR}}(\mathrm{PT}_{\nabla}[M, g, f]). $$ + +Using (5.96) and (5.98), we have + +$$ +\begin{align*} +\mathrm{PT}_{\mathrm{sing},\infty}(h_{\mathrm{sing},\infty})([M, g, t, f]) &= h_{\mathrm{sing},\infty}(\mathrm{PT}_{\nabla,\tau}[M, g, t, f]) \\ +&= h_{\mathrm{dR}}(\mathrm{PT}_{\nabla}[M, g, f]) - \langle B_{g_{f*}\nabla}((f|_{M_f})^*\mathrm{pr}_X^*\omega), t_{M_f,\tau}\rangle_{M_f}. +\end{align*} +$$ + +So it is enough to show that + +$$ (5.104) \qquad +\begin{aligned} +& \left< f^* \left( B(\pi^* \mathrm{cw}_{\nabla}^{\mathrm{dR}}(\mathrm{pr}_X^*\omega), \mathrm{pr}_{D(W_q)}^*\tau) + \pi^* B_\nabla(\mathrm{pr}_X^*\omega) \cup \mathrm{pr}_{D(W_q)}^*\tau \right), t_M \right> \\ +& = \langle B_{g_{f*}\nabla}((f|_{M_f})^*\mathrm{pr}_X^*\omega), t_{M_f,\tau} \rangle_{M_f}. +\end{aligned} +$$ +---PAGE_BREAK--- + +Recall that we have $\nu = f^{-1}(X \times \mathring{\tilde{D}}(\mathcal{W}_q))$ and a decomposition $t = t_{\bar{\nu}}+(t-t_{\bar{\nu}})$ so that $t_{\bar{\nu}} \in C_{\infty,n-1}(\bar{\nu}; \text{Ori}(M))$ and $t - t_{\bar{\nu}} \in C_{\infty,n-1}(M \setminus \nu; \text{Ori}(M))$. Also using the naturality of $B$, we have + +$$ +\begin{aligned} +& \left\langle f^* \left( B(\pi^* \mathrm{cw}_{\bar{\nabla}}^{\mathrm{dR}}(\mathrm{pr}_X^*\omega), \mathrm{pr}_{D(W_q)}^*\tau \right) + \pi^* B_\nabla (\mathrm{pr}_X^*\omega) \cup \mathrm{pr}_{D(W_q)}^*\tau \right), t \right\rangle_M \\ +&= \left\langle B(f^*\pi^*\mathrm{cw}_{\bar{\nabla}}^{\mathrm{dR}}(\mathrm{pr}_X^*\omega), f^*\mathrm{pr}_{D(W_q)}^*\tau), t_{\bar{\nu}} \right\rangle_{\bar{\nu}} + \left\langle f^* \left( \pi^* B_\nabla (\mathrm{pr}_X^*\omega) \cup \mathrm{pr}_{D(W_q)}^*\tau \right), t_{\bar{\nu}} \right\rangle_{\bar{\nu}}. +\end{aligned} +$$ + +We claim that the first term in (5.105) is zero. Indeed, by (3.16) we have +$f|_{M_f} \circ \pi_f = \pi \circ f: \nu \to X \times B$, so we have + +$$f^* \pi^* \mathrm{cw}_{\bar{\nabla}}^{\mathrm{dR}}(\mathrm{pr}_X^* \omega) = \pi_f^* (f|_{M_f})^* \mathrm{cw}_{\bar{\nabla}}^{\mathrm{dR}}(\mathrm{pr}_X^* \omega).$$ + +Recall that we have $\mathrm{cw}_{\bar{\nabla}}^{\mathrm{dR}}(\mathrm{pr}_X^*\omega) \in \Omega_{\mathrm{clo}}^n(X \times B; \mathrm{Ori}(\pi))$. Since the dimension of $M_f$ is $(n-1)$, we see that $(f|_{M_f})^*\mathrm{cw}_{\bar{\nabla}}^{\mathrm{dR}}(\mathrm{pr}_X^*\omega) = 0$. Thus we see that the first term of (5.105) is zero. On the other hand, the naturality of $B$ implies that + +$$B_{g_f * \nabla} ((f|_{M_f})^* \mathrm{pr}_X^* \omega) = (f|_{M_f})^* B_\nabla (\mathrm{pr}_X^* \omega).$$ + +We have + +$$ +\begin{align*} +\langle B_{g_{f_*}\nabla}((f|_{M_f})^* \mathrm{pr}_X^*\omega), t_{M_f, \tau} \rangle_{M_f} &= \langle (f|_{M_f})^* B_\nabla(\mathrm{pr}_X^*\omega), t_{M_f, \tau} \rangle_{M_f} \\ +&= \left\langle \pi_f^*(f|_{M_f})^* B_\nabla(\mathrm{pr}_X^*\omega), (\mathrm{pr}_{D(W_q)} \circ f)^*\tau \cap t_{\bar{\nu}} \right\rangle_{\bar{\nu}} \\ +&= \left\langle \pi_f^*(f|_{M_f})^* B_\nabla(\mathrm{pr}_X^*\omega) \cup (\mathrm{pr}_{D(W_q)} \circ f)^*\tau, t_{\bar{\nu}} \right\rangle_{\bar{\nu}} \\ +&= \left\langle f^*\left(\pi^* B_\nabla(\mathrm{pr}_X^*\omega) \cup \mathrm{pr}_{D(W_q)}^*\tau\right), t_{\bar{\nu}} \right\rangle_{\bar{\nu}}. +\end{align*} +$$ + +where the second equality used (5.103) and the last used $f|_{M_f} \circ \pi_f = \pi \circ f$ +again. This proves the equality (5.104) and completes the proof. $\square$ + +The construction in this subsection is particularly useful in the follow- +ing setting. Fix a positive integer $N$. In many examples of $G$, there exists a +positive integer $d$ so that the map $\rho_{d'} : BG_{d'} \to BG_{d'+1}$ is $(N+1)$-connected +for all $d' \ge d$. Fix such $d$, and take $P$, $B$ and $\mathcal{W}_q$ with $q \ge 2$ such that the +classifying map $B \to BG_d$ is $(N+1)$-connected. This means that the map +(5.90) is $(N+q+1)$-connected, so by Lemma 5.29 we have the isomorphism + +$$ (5.105) \qquad f^* \circ \text{susp}^q : IZ^n (- \wedge MTG) \simeq IZ^{n+q} (- \times \text{Thom}_\infty(\mathcal{W}_q)) $$ + +for $n \le N$. Thus, Proposition 5.92 implies that, if we are interested in +degree * $\le N$, the transformation $\mathcal{P}_{d_R,P}$ is an isomorphism and equivalent +to the transformation $\mathcal{P}$ up to sign, restricted to manifolds. + +6. EXAMPLES REVISITED + +In this subsection, we analyze some of the examples given in Subsection +4.2. We characterize the elements in $I\Omega_{dR}^G$ in a homotopy-theoretic way. +First we look into the holonomy theory (1) in Example 4.79. + +**Proposition 6.1.** The element $I(c_1(\nabla), \text{Hol}_\nabla) \in (I\Omega_{dR}^{\text{SO}})^2(X) = [X^+ \wedge MTSO, \Sigma^2 IZ]$ in Example 4.79 coincides with the following composition, + +$$ X^+ \wedge MTSO \xrightarrow[c_1(L)\wedge\tau]{c_1(L)\wedge\tau} \Sigma^2 HZ \wedge HZ \xrightarrow[\text{multi}]{\text{multi}} \Sigma^2 HZ \xrightarrow{\gamma} \Sigma^2 IZ. $$ +---PAGE_BREAK--- + +Here $\tau \in [\text{MTSO}, \mathbb{H}\mathbb{Z}]$ is the universal Thom class. + +*Proof.* The statement is equivalent to the claim that the image of the element $c_1(L) \cup \tau \in \mathbb{H}\mathbb{Z}^2(X^+ \wedge MTSO)$ under the self-duality homomorphism $\gamma: \mathbb{H}\mathbb{Z}^* \to \mathbb{I}\mathbb{Z}^*$ coincides with $\mathcal{P} \circ I(c_1(\nabla), \text{Hol}_{\nabla})$, where $\mathcal{P}: (\mathcal{I}\Omega^{\text{SO}})^* \simeq \mathbb{I}\mathbb{Z}^*(-\wedge MTSO)$ is given in Definition 5.50. + +Recall that, by Proposition 5.92 and the discussion following it, the transformation $\mathcal{P}$ can be described in the differential model, using an approximation of BSO by manifolds. Take a principal SO$(d)$-bundle $P \to B$ over a manifold $B$ and a rank-$q$ vector bundle $W_q \to B$ with a fixed isomorphism (5.83) such that the classifying map $B \to BSO$ is 3-connected. Take a connection $\nabla_P$ on $P$ and a Thom form $\tau \in \Omega_{clo}^q(\text{Thom}_\infty(W_q); \text{Ori}(\pi))$. This represents the pullback of the universal Thom class. By Definition 5.84 and Proposition 5.92, a differential lift of the element $\mathcal{P} \circ I(c_1(\nabla), \text{Hol}_{\nabla})$ is given by (note that $cw_{\nabla_P}(pr_X^*c_1(\nabla)) = pr_X^*c_1(\nabla)$ in this case), + +$$ (6.2) \quad \hat{P}_{\mathrm{dR},\nabla_P,\tau}(c_1(\nabla), \mathrm{Hol}_\nabla) = (\pi^*\mathrm{pr}_X^*c_1(\nabla) \wedge \mathrm{pr}_{D(W_q)}^\tau, \mathrm{PT}_{\nabla_P,\tau}(\mathrm{Hol}_\nabla)) \in \widehat{\mathcal{I}\mathbb{Z}}_{\mathrm{dR}}^{2+q}(X \times \mathrm{Thom}_\infty(W_q)). $$ + +It is enough to show that it is the image of an element of Cheeger-Simons differential character group $\hat{H}_{\mathrm{CS}}^{2+q}(X \times \mathrm{Thom}_\infty(W_q); \mathbb{Z})$ under the differential lift $\hat{\gamma}_{\mathrm{dR}}$ of the self-duality homomorphism (Definition 4.136). + +First, notice that $(c_1(L), \mathrm{Hol}_\nabla)$ gives a Cheeger-Simons differential character which we denote as $\mathcal{L} \in \hat{H}_{\mathrm{CS}}^2(X; \mathbb{Z})$. Next, notice that $\mathrm{Thom}(W_q)$ is $(q-1)$-connected and hence a choice of a Thom form $\tau \in \Omega_{clo}^q(\mathrm{Thom}_\infty(W_q); \mathrm{Ori}(\pi))$ lifts uniquely to an element $\mathcal{T} \in \hat{H}_{\mathrm{CS}}^q(\mathrm{Thom}_\infty(W_q); \mathbb{Z})$. We have a product operation in $\hat{H}_{\mathrm{CS}}$ ([CS85]) which we denote by $\star$. Using it, we get + +$$ (\mathrm{pr}_X^* \mathcal{L}) \star (\mathrm{pr}_{D(W_q)}^\tau \mathcal{T}) \in \hat{H}_{\mathrm{CS}}^{2+q}(X \times \mathrm{Thom}_\infty(W_q); \mathbb{Z}) $$ + +which lifts $\mathrm{pr}_X^* c_1(L) \cup \mathrm{pr}_{D(W_q)}^\tau \in H^{2+q}(X \times \mathrm{Thom}_\infty(W_q); \mathbb{Z})$. We check that its image under $\hat{\gamma}_{\mathrm{dR}}$ coincides with $\hat{P}_{\mathrm{dR},\nabla_P,\tau}(c_1(\nabla), \mathrm{Hol}_\nabla)$. Note that the differential form parts coincide, since $R(x \star y) = R(x) \wedge R(y)$ in general. Thus we only need to evaluate it on nice elements of the geometric stable tangential fr-cycles $C_{\infty,q+1}^{\mathrm{fr}}(X \times \mathrm{Thom}_\infty(W_q))$ since more general case follows from the compatibility condition. Let $(M,g,f)$ be nice with a tubular neighborhood structure $\pi_f: \nu \to M_f$. We have + +$$ f^*(\mathrm{pr}_X^* \mathcal{L}) = \pi_f^*((\mathrm{pr}_X \circ f|_{M_f})^* \mathcal{L}) $$ + +which follows from (3.16). Now $(\mathrm{pr}_X \circ f|_{M_f})^* \mathcal{L}$ is an element of $\hat{H}_{\mathrm{CS}}^2(M_f; \mathbb{Z})$. Since $H^2(M_f, \mathbb{Z}) = 0$ for the one-dimensional manifold $M_f$, there exists an element $\alpha \in \Omega^1(M_f)$ such that $(\mathrm{pr}_X \circ f|_{M_f})^* \mathcal{L} = a_{\mathrm{CS}}(\alpha)$. By using the property of the product $\star$ that $a_{\mathrm{CS}}(\alpha) \star (\omega, k) = a_{\mathrm{CS}}(\alpha \wedge \omega)$ ([CS85, (1.15)]), we get + +$$ +\begin{align*} +f^*(\mathrm{pr}_X^*\mathcal{L}) * f^*(\mathrm{pr}_{D(W_q)}^\tau\mathcal{T}) &= a_{\mathrm{CS}}(\pi_f^*\alpha) * (\mathrm{pr}_{D(W_q)} \circ f)^*\mathcal{T} \\ +&= a_{\mathrm{CS}}(\pi_f^*\alpha \wedge (\mathrm{pr}_{D(W_q)} \circ f)^*\tau). +\end{align*} +$$ +---PAGE_BREAK--- + +By evaluating it on $M$, we get + +$$ \int_{\nu} \pi_{f}^{*}\alpha \wedge (\mathrm{pr}_{D(W_{q})} \circ f)^{*}\tau = \int_{M_{f}} \alpha $$ + +The left hand side is the result of evaluation of $\hat{\gamma}_{dR}((\mathrm{pr}_X^*\mathcal{L}) \star (\mathrm{pr}_D^*(W_q^\mathcal{T})))$, +while the right hand side is that of $\hat{P}_{dR,\nabla_P,\tau}(c_1(\nabla), \mathrm{Hol}\nabla)$. Therefore, we +get the desired result. + +Next we look at the classical Chern-Simons theory in Example 4.81. + +**Proposition 6.3.** The element $I(1 \otimes \lambda_{\mathbb{R}}, h_{CS_{\hat{\lambda}}}) \in (I\Omega_{dR}^{\text{SO}\times H})^n(\text{pt})$ given in (4.86) coincides with the following composition, + +$$ BH^+ \wedge \mathrm{MTSO} \xrightarrow{\lambda \wedge \tau} \Sigma^n HZ \wedge HZ \xrightarrow{\text{multi}} \Sigma^n HZ \xrightarrow{\gamma} \Sigma^n IZ. $$ + +Here $\tau \in [\mathrm{MTSO}, HZ]$ is the universal Thom class. + +*Proof.* The proof is similar to that of Proposition 6.1 above. We check that the image of the element $\lambda \cup \tau \in HZ^n(BH^+ \wedge \mathrm{MTSO})$ under the self-duality homomorphism $\gamma: HZ^* \to IZ^*$ coincides with $\mathcal{P} \circ I(1 \otimes \lambda_{\mathbb{R}}, h_{CS_{\hat{\lambda}}})$, where $\mathcal{P}: (I\Omega^{\text{SO}\times H})^* \simeq IZ^*(-\wedge BH^+ \wedge \mathrm{MTSO})$ is given in Definition 5.50. + +Take $d$ large enough, and take a principal SO($d, \mathbb{R}$)-bundle $E_{n+1}\mathrm{SO} \to B_{n+1}\mathrm{SO}$ over a manifold such that the classifying map $B_{n+1}\mathrm{SO} \to B\mathrm{SO}$ is $(n+1)$-connected. Take a vector bundle $\pi: W_q \to B_{n+1}\mathrm{SO}$ as in the proof of Proposition 6.1, a connection $\nabla_{\mathrm{SO}}$ on $E_{n+1}\mathrm{SO}$ and a Thom form $\tau$ for $W_q$. Also take a principal $H$-bundle $E_{n+1}H \to B_{n+1}H$ over a manifold such that the classifying map $B_{n+1}H \to BH$ is $(n+1)$-connected, and take a connection $\nabla_H$ on it. Note that, since the homomorphism $\rho: \mathrm{SO} \times H \to \mathrm{O}$ is given by crashing $H$, the Thom space of the pullback of $W_q$ to $B_{n+1}H \times B_{n+1}\mathrm{SO}$ plays the role of an approximation of $\mathrm{MT}(\mathrm{SO} \times H) = BH^+ \wedge \mathrm{MTSO}$. + +By Definition 5.84 and Proposition 5.92, a differential lift of the element $\mathcal{P} \circ I(1 \otimes \lambda_{\mathbb{R}}, h_{CS_{\hat{\lambda}}})$ is given by + +$$ (6.4) \quad \hat{P}_{dR,(\nabla_{\mathrm{SO}}+\nabla_H),\tau}(1 \otimes \lambda_{\mathbb{R}}, h_{CS_{\hat{\lambda}}}) = (\mathrm{pr}_{B_{n+1}H}^* c w_{\nabla_H}(\lambda_{\mathbb{R}}) \wedge \tau, P T_{\nabla_{\mathrm{SO}}+\nabla_H,\tau}(h_{CS_{\lambda}})) \\ \in \widehat{I\mathbb{Z}}_{dR}^{n+q}(B_{n+1}H \times \mathrm{Thom}_{\infty}(W_q)) $$ + +By the same argument as in the proof of Proposition 6.1, we see that +the element (6.4) coincides with the image of a differential lift of $\lambda \cup \tau \in$ +$HZ^n(BH^+ \wedge MTSO)$ under the differential lift $\hat{\gamma}_{dR}$ of the self-duality ho- +momorphism (Definition 4.136). (We use $H^n(M_f; \mathbb{Z}) = 0$ on the $(n-1)$- +manifold $M_f$. The details are left to the reader.) Thus we get the result. +$\square$ + +*Remark 6.5.* It is natural to expect similar characterizations of other ex- +amples given in Subsection 4.2. For example the element $I((\text{Todd})|_{2k}, \bar{\eta}) \in$ +$(I\Omega_{dR}^{\text{Spin}})^{2k}(\text{pt})$ appearing in Example 4.87 is expected to coincide with + +$$ \mathrm{MT}\mathrm{Spin}^c \xrightarrow{\mathrm{ABS}} K \xrightarrow{\mathrm{Bott}} \Sigma^{2k} K \xrightarrow{\gamma_K} \Sigma^{2k} I\mathbb{Z}, $$ +---PAGE_BREAK--- + +where the first map is the Atiyah-Bott-Shapiro orientation, the second is +the Bott periodicity and the third is the Anderson self-duality element for +K-theory. Actually we can show that it is indeed true. The key point is +that the reduced eta invariant $\bar{\eta}$ gives the odd-dimensional pushforward in +differential K-theory by the result of Freed and Lott [FL10]. The authors +plan to develop a unified treatment of such examples, in particular a proof +of the above claim, in a subsequent paper. + +7. CONCLUDING REMARKS + +**7.1. Normal G-structures.** In this paper we have focused on the *tangential* $G$-bordism theories and its Anderson duals. However, by a straightforward modification, we can construct the corresponding models for the Anderson duals $(I\Omega^{G^\perp})^*$ to the *normal* $G$-bordism theories $\Omega^{G^\perp}$. In this subsection we outline the construction. + +First we need to introduce *geometric stable normal* $G$-structures on manifolds. A *stable normal bundle* on a manifold $M$ consists of a choice of an injective homomorphism of vector bundles + +$$ \alpha: TM \hookrightarrow \mathbb{R}^k $$ + +over $M$. Given such an $\alpha$, the associated *stable normal bundle* on $M$ is +defined to be the cokernel of $\alpha$. A *geometric stable normal G-structure* +$g$ on a manifold $M$ is a choice of $\alpha$ as above together with a geometric +stable $G$-structure (Definition 3.1) on the associated stable normal bundle. +For a $\langle k \rangle$-manifolds $M$ with collar structures, we require the compatibility +with the collar structures as before. Then for a pair of manifolds $(X, A)$, +a *geometric smooth stable normal G-cycle* of dimension $n$ over $(X, A)$ is +a triple $(M, g, f)$ as in Definition 3.7, where now $g$ is a geometric stable +normal $G$-structure on $M$. The corresponding abelian group $C_{\infty,n}^{G^\perp}(X, A)$ +consists of the equivalence classes of such $(M, g, f)$ under the equivalence +relation given by isomorphisms, stabilizations and opposite relations. We +also define bordism data for geometric smooth stable normal $G$-cycles as in +Definition 3.8. + +The variant of the Chern-Weil construction in Definition 4.4 also applies +to the normal settings. First note that we have + +$$ (I\Omega^{G^\perp})^*(pt) \otimes \mathbb{R} = \varinjlim_d H^*(G_d; \mathbb{R}^{G_d}). $$ + +The proof is the same as that of Lemma 4.1, where now the Madsen-Tillmann +spectrum *MTG* is replaced by the Thom spectrum *MG*. By Lemma 4.2 we +denote (see (2.56)) + +$$ V_{I\Omega^{G\perp}}^{\bullet} := \varprojlim_{d} H^{\bullet}(BG_d; \mathbb{R}^{G_d}) = \varprojlim_{d} (\mathrm{Sym}^{\bullet/2}_{1/2} g_d^* \otimes_{\mathbb{R}} \mathbb{R}^{G_d})^{G_d}. $$ + +Note that we have $V_{I\Omega^{G\perp}}^{\bullet} = V_{I\Omega^{G}}^{\bullet}$. + +Suppose we are given a manifold $W$ with a geometric stable normal $G$ +structure $g$ with underlying stable normal bundle given by $\alpha: TW \hookrightarrow \mathbb{R}^k$. +Proceeding as in Definition 4.4, given $\phi \in V_{I\Omega^{G\perp}}^n$ we get + +$$ (7.1) \qquad \operatorname{cw}_g(\phi) \in \Omega_{\operatorname{clo}}^n(W; \operatorname{Ori}(\operatorname{coker}(\alpha))). $$ +---PAGE_BREAK--- + +Using the isomorphism $\text{Ori}(\text{coker}(\alpha))) \simeq \text{Ori}(W)$ given by $TW \oplus \text{coker}(\alpha) \simeq \mathbb{R}^k$ and our sign convention (2.35) and (2.36), we regard + +$$ \mathrm{cw}_g(\phi) \in \Omega_{\mathrm{clo}}^n(W; \mathrm{Ori}(W)). $$ + +Then as in (4.9) we get a homomorphism + +$$ (7.2) \qquad \mathrm{cw}_g : \Omega^*(W; V_{I\Omega^{G^\perp}}^\bullet) \to \Omega^*(W; \mathrm{Ori}(W)). $$ + +Now we can define the differential model $(I\Omega_{\mathrm{dR}}^{G^\perp})^*$ and its differential extension $(\widehat{I\Omega_{\mathrm{dR}}^{G^\perp}})^*$ as follows. + +**Definition 7.3** ($\widehat{(I\Omega_{\mathrm{dR}}^{G^\perp})}^*$ and $(I\Omega_{\mathrm{dR}}^{G^\perp})^*$). Let $(X, A)$ be a pair of manifolds and $n \in \mathbb{Z}_{\ge 0}$. + +(1) Define $(\widehat{I\Omega_{\mathrm{dR}}^{G^\perp}})^n(X, A)$ to be an abelian group consisting of pairs $(\omega, h)$, such that + +(a) $\omega$ is a closed $n$-form $\omega \in \Omega_{\mathrm{clo}}^n(X, A; V_{I\Omega^{G^\perp}}^\bullet)$. + +(b) $h$ is a group homomorphism $h: C_{\infty, n-1}^{G^\perp}(X, A) \to \mathbb{R}/\mathbb{Z}$. + +(c) $\omega$ and $h$ satisfy the following compatibility condition. Assume that we are given a bordism data $(W, g_W, f_W)$ of a geometric smooth stable normal $G$-cycle $(M, g_M, f_M)$ of dimension $(n-1)$ over $(X, A)$. Then we have + +$$ h[M, g_M, f_M] = \int_W \mathrm{cw}_{g_W}(f_W^*\omega) \pmod{\mathbb{Z}}. $$ + +Abelian group structure is defined in the obvious way. + +(2) We define a homomorphsim of abelian groups, + +$$ (7.4) \qquad \begin{aligned} a: & \Omega^{n-1}(X, A; V_{I\Omega^{G^\perp}}^\bullet)/\operatorname{Im}(d) \to (\widehat{I\Omega_{\mathrm{dR}}^{G^\perp}})^n(X, A) \\ & \alpha \mapsto (d\alpha, \mathrm{cw}(\alpha)). \end{aligned} $$ + +Here the homomorphism $\mathrm{cw}(\alpha): C_{\infty, n-1}^{G^\perp}(X, A) \to \mathbb{R}/\mathbb{Z}$ is defined by + +$$ (7.5) \qquad \mathrm{cw}(\alpha)([M, g, f]) := \int_M \mathrm{cw}_g(f^*\alpha) \pmod{\mathbb{Z}}. $$ + +We set + +$$ (I\Omega_{\mathrm{dR}}^{G^\perp})^n(X, A) := (\widehat{I\Omega_{\mathrm{dR}}^{G^\perp}})^n(X, A)/\operatorname{Im}(a). $$ + +For $n \in \mathbb{Z}_{<0}$ we set $(\widehat{I\Omega_{\mathrm{dR}}^{G^\perp}})^n(X, A) = 0$ and $(I\Omega_{\mathrm{dR}}^{G^\perp})^n(X, A) = 0$. + +We can prove that $(I\Omega_{\mathrm{dR}}^{G^\perp})^*$ is indeed a model for $(I\Omega^{G^\perp})^*$, also by modifying the argument in Section 5. The structure homomorphisms $(R, a, I)$ for $(\widehat{I\Omega_{\mathrm{dR}}^{G^\perp}})^*$ are constructed in the same way as Definition 4.15, making it into a differential extension of $(I\Omega^{G^\perp})^*$. +---PAGE_BREAK--- + +7.2. **Module structures.** Bunke, Schick, Schröder and Wiethaup [BSSW09] gave a model for a differential extension of $(\Omega^{G^\perp})^*$ (normal $G$-bordisms). They provide the detail for the case of complex bordisms, but as they note, their construction directly generalizes to any $G$. Moreover, their construction can be modified to give a model for $(\Omega^G)^*$ (tangential $G$-bordisms). The relation can be briefly explained as follows. + +In general, for a ring spectrum $E$, its Anderson dual $IE$ is a $E$-module spectrum, i.e., we have a multiplication map + +$$ (7.6) \qquad E^k(X) \otimes IE^n(X) \to IE^{n+k}(X) $$ + +for any CW complex $X$ natural in $X$. Suppose $G$ is *multiplicative*, in the sense that it is equipped with group homomorphisms $G_n \times G_m \to G_{n+m}$ which are compatible with the structure homomorphisms in $G$. Then *MTG* becomes multiplicative, so we get the $(\Omega^G)^{-}$-module structure on $(I\Omega^G)^*$, + +$$ (7.7) \qquad (\Omega^G)^{-r}(X) \otimes (I\Omega^G)^n(X) \to (I\Omega^G)^{n-r}(X). $$ + +In [BSSW09] they provide the model for a differential extension of $(\Omega^{G^\perp})^*$ in terms of “geometric cycles”, which turns out to be very much suited to the constructions in this paper. We can actually construct a differential refinement of the module structure (7.7) using their models and our models. The details will appear in a subsequent paper. + +## ACKNOWLEDGMENT + +The authors are grateful to Yosuke Morita for collaboration on many parts of the paper, and Kaoru Ono for helpful discussion and comments. The work of MY is supported by Grant-in-Aid for JSPS KAKENHI Grant Number 20K14307 and JST CREST program JPMJCR18T6. The work of KY is supported by JST FOREST Program (Grant Number JPMJFR2030, Japan), and JSPS KAKENHI Grant Number 17K14265. + +## REFERENCES + +[APS75a] M. F. Atiyah, V. K. Patodi, and I. M. Singer, *Spectral asymmetry and Riemannian geometry*. II, Math. Proc. Cambridge Philos. Soc. **78** (1975) 405–432. + +[APS75b] ————, *Spectral asymmetry and Riemannian geometry*. I, Math. Proc. Cambridge Philos. Soc. **77** (1975) 43–69. + +[APS76] ————, *Spectral asymmetry and Riemannian geometry*. III, Math. Proc. Cambridge Philos. Soc. **79** (1976) 71–99. + +[BS10] U. Bunke and T. Schick, *Uniqueness of smooth extensions of generalized cohomology theories*, J. Topol. **3** (2010) 110–156. + +[BSSW09] U. Bunke, T. Schick, I. Schröder, and M. Wiethaup, *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751–1790. + +[BT06] M. Brightwell and P. Turner, *Relative differential characters*, Comm. Anal. Geom. **14** (2006) 269–282. + +[CFLS19a] C. Córdova, D. S. Freed, H. T. Lam, and N. Seiberg, *Anomalies in the Space of Coupling Constants and Their Dynamical Applications*. I, SciPost Phys. **8** (2020) 001, arXiv:1905.09315 [hep-th]. + +[CFLS19b] ————, *Anomalies in the Space of Coupling Constants and Their Dynamical Applications*. II, SciPost Phys. **8** (2020) 002, arXiv:1905.13361 [hep-th]. + +[CL20] C.-M. Chang and Y.-H. Lin, *On Exotic Consistent Anomalies in (1+1)d: A Ghost Story*, arXiv:2009.07273 [hep-th]. +---PAGE_BREAK--- + +[CS85] J. Cheeger and J. Simons, *Differential characters and geometric invariants*, Geometry and topology (College Park, Md., 1983/84), Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 50–80. https://doi.org/10.1007/BFb0075216. + +[DL20] J. Davighi and N. Lohitsiri, *The algebra of anomaly interplay*, SciPost Phys. **10** (2021) 074, arXiv:2011.10102 [hep-th]. + +[FH21] D. S. Freed and M. J. Hopkins, *Reflection positivity and invertible topological phases*, Geom. Topol. **25** (2021) 1165–1330. + +[FL10] D. S. Freed and J. Lott, *An index theorem in differential K-theory*, Geom. Topol. **14** (2010) 903–966. + +[FMS07] D. S. Freed, G. W. Moore, and G. Segal, *The uncertainty of fluxes*, Comm. Math. Phys. **271** (2007) 247–274. + +[Fre95] D. S. Freed, *Classical Chern-Simons theory. I*, Adv. Math. **113** (1995) 237–303. + +[Fre02] __________, *Classical Chern-Simons theory. II*, vol. 28, 2002, pp. 293–310. Special issue for S. S. Chern. + +[Fre14] __________, *Anomalies and Invertible Field Theories*, Proc. Symp. Pure Math. **88** (2014) 25–46, arXiv:1404.7224 [hep-th]. + +[Fre19] __________, *Lectures on field theory and topology*, CBMS Regional Conference Series in Mathematics, vol. 133, American Mathematical Society, Providence, RI, 2019. https://doi.org/10.1090/cbms/133. Published for the Conference Board of the Mathematical Sciences. + +[Gug76] V. K. A. M. Gugenheim, *On the multiplicative structure of the de Rham theory*, J. Differential Geometry **11** (1976) 309–314. + +[Hat01] A. Hatcher, *Algebraic topology*, Cambridge University Press, 2001. + +[HKT20] P.-S. Hsin, A. Kapustin, and R. Thorngren, *Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena*, Phys. Rev. B **102** (2020) 245113, arXiv:2004.10758 [cond-mat.str-el]. + +[HS05] M. J. Hopkins and I. M. Singer, *Quadratic functions in geometry, topology, and M-theory*, J. Differential Geom. **70** (2005) 329–452. + +[HTY19] C.-T. Hsieh, Y. Tachikawa, and K. Yonekura, *Anomaly of the Electromagnetic Duality of Maxwell Theory*, Phys. Rev. Lett. **123** (2019) 161601, arXiv:1905.08943 [hep-th]. + +[HTY20] __________, *Anomaly inflow and p-form gauge theories*, arXiv:2003.11550 [hep-th]. + +[J68] K. Jänich, *On the classification of O(n)-manifolds*, Math. Ann. **176** (1968) 53–76. + +[Kap14] A. Kapustin, *Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology*, arXiv:1403.1467 [cond-mat.str-el]. + +[KTTW14] A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, *Fermionic Symmetry Protected Topological Phases and Cobordisms*, JHEP **12** (2015) 052, arXiv:1406.7329 [cond-mat.str-el]. + +[LMSN18] C. Lawrie, D. Martelli, and S. Schäfer-Nameki, *Theories of Class F and Anomalies*, JHEP **10** (2018) 090, arXiv:1806.06066 [hep-th]. + +[LOT20] Y. Lee, K. Ohmori, and Y. Tachikawa, *Revisiting Wess-Zumino-Witten terms*, SciPost Phys. **10** (2021) 061, arXiv:2009.00033 [hep-th]. + +[MN84] G. W. Moore and P. C. Nelson, *Anomalies in Nonlinear σ Models*, Phys. Rev. Lett. **53** (1984) 1519. + +[MN85] __________, *The Etiology of σ Model Anomalies*, Commun. Math. Phys. **100** (1985) 83. + +[Mon19] S. Monnier, *A Modern Point of View on Anomalies*, Fortsch. Phys. **67** (2019) 191–282, arXiv:1903.02828 [hep-th]. + +[NR61] M. S. Narasimhan and S. Ramanan, *Existence of universal connections*, Amer. J. Math. **83** (1961) 563–572. + +[Qui67] D. G. Quillen, *Homotopical algebra*, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. +---PAGE_BREAK--- + +[Ros94] J. Rosenberg, *Algebraic K-theory and its applications*, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4612-4314-4. + +[Rud98] Y. B. Rudyak, *On Thom spectra, orientability, and cobordism*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. With a foreword by Haynes Miller. + +[ST19] A. Schwimmer and S. Theisen, *Osborn Equation and Irrelevant Operators*, J. Stat. Mech. 1908 (2019) 084011, arXiv:1902.04473 [hep-th]. + +[STY18] N. Seiberg, Y. Tachikawa, and K. Yonekura, *Anomalies of Duality Groups and Extended Conformal Manifolds*, PTEP 2018 (2018) 073B04, arXiv:1803.07366 [hep-th]. + +[Tho17] R. Thorngren, *Topological Terms and Phases of Sigma Models*, arXiv:1710.02545 [cond-mat.str-el]. + +[TY17] Y. Tachikawa and K. Yonekura, *Anomalies involving the space of couplings and the Zamolodchikov metric*, JHEP 12 (2017) 140, arXiv:1710.03934 [hep-th]. + +[Wit83] E. Witten, *Global Aspects of Current Algebra*, Nucl. Phys. B 223 (1983) 422-432. + +[WY19] E. Witten and K. Yonekura, *Anomaly Inflow and the η-Invariant*, The Shoucheng Zhang Memorial Workshop, 9 2019. arXiv:1909.08775 [hep-th]. + +[WZ71] J. Wess and B. Zumino, *Consequences of anomalous Ward identities*, Phys. Lett. B 37 (1971) 95-97. + +[Yon18] K. Yonekura, *On the cobordism classification of symmetry protected topological phases*, Commun. Math. Phys. 368 (2019) 1121-1173, arXiv:1803.10796 [hep-th]. \ No newline at end of file