# B-Bordism

## 1 Introduction

On this page we recall the definition of the bordism groups of closed smooth manifolds, with extra topological structure: orientation, spin-structure, weakly almost complex structure etc. The situation for piecewise linear and topological manifolds is similar and we discuss it briefly below.

The formulation of the general set-up for B-Bordism dates back to [Lashof1963]. There are detailed treatments in [Stong1968, Chapter II] and [Bröcker&tom Dieck1970] as well as summaries in [Teichner1992, Part 1: 1], [Kreck1999, Section 1], [Kreck&Lück2005, 18.10]. See also the Wikipedia bordism page.

We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\gamma : B \to BO$ where $BO$$BO$ denotes the classifying space of the stable orthogonal group and $B$$B$ is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes $B$$B$ for the fibration $\gamma$$\gamma$. Speaking somewhat imprecisely (precise details are below) a $B$$B$-manifold is a compact manifold $M$$M$ together with a lift to $B$$B$ of a classifying map for the stable normal bundle of $M$$M$:

$\displaystyle \xymatrix{ & B \ar[d]^{\gamma} \\ W \ar[r]^{\nu_W} \ar[ur]^{\bar \nu} & BO.}$

The $n$$n$-dimensional $B$$B$-bordism group is defined to be the set of closed $B$$B$-manifolds modulo the relation of bordism via compact $B$$B$-manifolds. Addition is given by disjoint union and in fact for each $n \geq 0$$n \geq 0$ there is a group

$\displaystyle \Omega_n^B := \{ (M, \bar \nu) \}/\equiv.$

Alternative notations are $\Omega_n(B)$$\Omega_n(B)$ and also $\Omega_n^G$$\Omega_n^G$ when $(B \to BO) = (BG \to BO)$$(B \to BO) = (BG \to BO)$ for $G \to O$$G \to O$ a stable representation of a topological group $G$$G$. Details of the definition and some important theorems for computing $\Omega_n^B$$\Omega_n^B$ follow.

### 1.1 Examples

We list some fundamental examples with common notation and also indicate the fibration B.

• Unoriented bordism: $\mathcal{N}_*$$\mathcal{N}_*$; $B = (BO = BO)$$B = (BO = BO)$.
• Oriented bordism: $\Omega_*$$\Omega_*$, $\Omega_*^{SO}$$\Omega_*^{SO}$; $B = (BSO \to BO)$$B = (BSO \to BO)$.
• Spin bordism: $\Omega_*^{Spin}$$\Omega_*^{Spin}$; $B = (BSpin \to BO)$$B = (BSpin \to BO)$.
• Spin$c$$c$ bordism: $\Omega_*^{Spin^{c}}$$\Omega_*^{Spin^{c}}$; $B = (BSpin^{c} \to BO)$$B = (BSpin^{c} \to BO)$.
• String bodism : $\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$$\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$; $B = (BO\langle 8 \rangle \to BO)$$B = (BO\langle 8 \rangle \to BO)$.
• Complex bordism : $\Omega_*^U$$\Omega_*^U$; $B = (BU \to BO)$$B = (BU \to BO)$.
• Special unitary bordism : $\Omega_*^{SU}$$\Omega_*^{SU}$; $B = (BSU \to BO)$$B = (BSU \to BO)$.
• Framed bordism : $\Omega_*^{fr}$$\Omega_*^{fr}$; $B = (PBO \to BO)$$B = (PBO \to BO)$, the path space fibration.

## 2 B-structures and bordisms

In this section we give a compressed accont of parts of [Stong1968, Chapter II]. Let $G_{r, m}$$G_{r, m}$ denote the Grassmann manifold of unoriented $r$$r$-planes in $\Rr^m$$\Rr^m$ and let $BO(r) = \text{lim}_{m \to \infty} G_{r, m}$$BO(r) = \text{lim}_{m \to \infty} G_{r, m}$ be the infinite Grassmannian and fix a fibration $\gamma_r : B_r \to BO(r)$$\gamma_r : B_r \to BO(r)$.

Definition 2.1. Let $\xi: E \to X$$\xi: E \to X$ be a rank r vector bundle classified by $\xi : X \to BO(r)$$\xi : X \to BO(r)$. A $B_r$$B_r$-structure on $\xi$$\xi$ is a vertical homotopy class of maps $\bar \xi : X \to B_r$$\bar \xi : X \to B_r$ such that $\gamma_r \circ \bar \xi = \xi$$\gamma_r \circ \bar \xi = \xi$.

Note that if $\xi_0$$\xi_0$ and $\xi_1$$\xi_1$ are isomorphic vector bundles over $X$$X$ then the sets of $B_r$$B_r$-structures on each are in bijective equivalence. However $B_r$$B_r$-structures are defined on specific bundles, not isomorphism classes of bundles: a specific isomorphism, up to appropriate equivalence, is required to give a bijection between the sets of $B_r$$B_r$ structures. Happily this is the case for the normal bundle of an embedding as we now explain. Let $M$$M$ be a compact manifold and let $i_0 : M \to \Rr^{n+r}$$i_0 : M \to \Rr^{n+r}$ be an embedding. Equipping $\Rr^{n+r}$$\Rr^{n+r}$ with the standard metric, the normal bundle of $i_0$$i_0$ is a rank r vector bundle over $M$$M$ classified by its normal Gauss map $\nu(i_0) : M \to G_{r, n+r} \subset BO(r)$$\nu(i_0) : M \to G_{r, n+r} \subset BO(r)$. If $i_1$$i_1$ is another such embedding and $r >> n$$r >> n$, then $i_1$$i_1$ is regularly homotopic to $i_0$$i_0$ and all regular homotopies are regularly homotopic relative to their endpoints (see [Hirsch1959]). A regular homotopy $H$$H$ defines an isomorphism $\alpha_H :\nu(i_0) \cong \nu(i_1)$$\alpha_H :\nu(i_0) \cong \nu(i_1)$ and a regular homotopy of regular homotopies gives a homotopy between these isomorphisms. Taking care one proves the following

Lemma 2.2 [Stong1968, p 15]. For r sufficiently large, (depending only on n) there is a 1-1 correspondence between the set of $B_r$$B_r$ structures of the normal bundles of any two embeddings $i_0, i_1 : M \to \Rr^{n+r}$$i_0, i_1 : M \to \Rr^{n+r}$.

This lemma is one motivation for the useful but subtle notion of a fibred stable vector bundle.

Definition 2.3. A fibred stable vector bundle $B = (B_r, \gamma_r, g_r)$$B = (B_r, \gamma_r, g_r)$ consists of the following data: a sequence of fibrations $\gamma_r : B_r \to BO(r)$$\gamma_r : B_r \to BO(r)$ together with a sequence of maps $g_r : B_r \to B_{r+1}$$g_r : B_r \to B_{r+1}$ fitting into the following commutative diagram

$\displaystyle \xymatrix{ B_r \ar[r]^{g_r} \ar[d]^{\gamma_r} & B_{r+1} \ar[d]^{\gamma_{r+1}} \\ BO(r) \ar[r]^{j_r} & BO(r+1) }$

where $j_r$$j_r$ is the standard inclusion. We let $B = \text{lim}_{r \to \infty}(B_r)$$B = \text{lim}_{r \to \infty}(B_r)$.

Remark 2.4. A fibred stable vector bundle $B$$B$ gives rise to a stable vector bundle as defined in [Kreck&Lück2005, 18.10]. One defines $E_r \to B_r$$E_r \to B_r$ to be the pullback bundle $\gamma_r^*(EO(r))$$\gamma_r^*(EO(r))$ where $EO(r)$$EO(r)$ is the universal r-plane bundle over $BO(r)$$BO(r)$. The diagram above gives rise to bundle maps $\bar g_r : E_r \oplus \underline{\Rr} \to E_{r+1}$$\bar g_r : E_r \oplus \underline{\Rr} \to E_{r+1}$ covering the maps $g_r$$g_r$; where $\underline{\Rr}$$\underline{\Rr}$ denotes the trivial rank 1 bundle over $B_r$$B_r$.

Now a $B_r$$B_r$-structure on the normal bundle of an embedding $i: M \to \Rr^{n+r}$$i: M \to \Rr^{n+r}$ defines a unique $B_{r+1}$$B_{r+1}$-structure on the composition of $i$$i$ with the standard inclusion $\Rr^{n+r} \to \Rr^{n+r+1}$$\Rr^{n+r} \to \Rr^{n+r+1}$. Hence we can make the following

Definition 2.5 [Stong1968, p 15]. Let $B$$B$ be a fibred stable vector bundle. A $B$$B$-structure on $M$$M$ is an equivalence class of $B_r$$B_r$-structure on $M$$M$ where two such structures are equivalent if they become equivalent for r sufficiently large. A $B$$B$-manifold is a pair $(M, \bar \nu)$$(M, \bar \nu)$ where $M$$M$ is a compact manifold and $\bar \nu$$\bar \nu$ is a $B$$B$-structure on $M$$M$.

If $W$$W$ is a compact manifold with boundary $\partial W$$\partial W$ then by choosing the inward-pointing normal vector along $\partial W$$\partial W$, a $B$$B$-structure on $W$$W$ restricts to a $B$$B$-structure on $\partial W$$\partial W$. In particular, if $(M, \bar \nu_M)$$(M, \bar \nu_M)$ is a closed $B$$B$ manifold then $W = M \times [0, 1]$$W = M \times [0, 1]$ has a canonical $B$$B$-structure $\bar \nu_{M \times [0, 1]}$$\bar \nu_{M \times [0, 1]}$ which restricts to $(M, \bar \nu_M)$$(M, \bar \nu_M)$ on $M \times \{ 0 \}$$M \times \{ 0 \}$. The restriction of this $B$$B$-structure to $M \times \{ 1 \}$$M \times \{ 1 \}$ is denoted $-\bar \nu$$-\bar \nu$: by construction $(M \sqcup M, \bar \nu \sqcup - \bar \nu)$$(M \sqcup M, \bar \nu \sqcup - \bar \nu)$ is the boundary of $(M \times [0, 1], \bar \nu_{M \times [0, 1]})$$(M \times [0, 1], \bar \nu_{M \times [0, 1]})$.

Definition 2.6. Closed $B$$B$-manifolds $(M_0, \bar \nu_0)$$(M_0, \bar \nu_0)$ and $(M_1, \bar \nu_1)$$(M_1, \bar \nu_1)$ are $B$$B$-bordant if there is a compact $B$$B$-manifold $(W, \bar \nu)$$(W, \bar \nu)$ such that $\partial(W, \bar \nu) = (M_0 \sqcup M_1, \bar \nu_0 \sqcup -\bar \nu_1)$$\partial(W, \bar \nu) = (M_0 \sqcup M_1, \bar \nu_0 \sqcup -\bar \nu_1)$. We write $[M, \bar \nu]$$[M, \bar \nu]$ for the bordism class of $(M, \bar \nu)$$(M, \bar \nu)$.

Proposition 2.7 [Stong1968, p 17]. The set of $B$$B$-bordism classes of closed n-manifolds with $B$$B$-structure,

$\displaystyle \Omega_n^B := \{ [M, \bar \nu ] \},$

forms an abelian group under the operation of disjoint union with inverse $-[M,\bar \nu] = [M, -\bar \nu]$$-[M,\bar \nu] = [M, -\bar \nu]$.

## 3 Singular bordism

$B$$B$-bordism gives rise to a generalised homology theory. If $X$$X$ is a space then the $n$$n$-cycles of this homology theory are pairs

$\displaystyle ((M, \bar \nu),~ f: M \to X)$

where $(M, \bar \nu)$$(M, \bar \nu)$ is a closed $n$$n$-dimensional $B$$B$-manifold and $f$$f$ is any continuous map. Two cycles $((M_0, \bar \nu_0), f_0)$$((M_0, \bar \nu_0), f_0)$ and $((M_1, \bar \nu_1), f_1)$$((M_1, \bar \nu_1), f_1)$ are homologous if there is a pair

$\displaystyle ((W, \bar \nu),~ g : W \to X)$

where $(W, \bar \nu)$$(W, \bar \nu)$ is a $B$$B$-bordism from $(M_0, \bar \nu_0)$$(M_0, \bar \nu_0)$ to $(M_1, \bar \nu_1)$$(M_1, \bar \nu_1)$ and $g : W \to X$$g : W \to X$ is a continuous map extending $f_0 \sqcup f_1$$f_0 \sqcup f_1$. Writing $[(M, \bar \nu), f]$$[(M, \bar \nu), f]$ for the equivalence class of $((M, \bar \nu) ,f)$$((M, \bar \nu) ,f)$ we obtain an abelian group

$\displaystyle \Omega_n^B(X) : = \{ [(M, \bar \nu), f] \}$

with group operation disjoint union and inverse $-[(M, \bar \nu), f] = [(M, - \bar \nu), f]$$-[(M, \bar \nu), f] = [(M, - \bar \nu), f]$.

Proposition 3.1. The mapping $X \to \Omega_n^B(X)$$X \to \Omega_n^B(X)$ defines a generalised homology theory with coefficients $\Omega_n^B(\text{pt}) = \Omega_n^B$$\Omega_n^B(\text{pt}) = \Omega_n^B$.

Given a stable vector bundle $B = (B_r, \gamma_r, g_r)$$B = (B_r, \gamma_r, g_r)$ we can form the stable vector bundle $B \times X := (B_r \times X, \gamma_r \times X, g_r \times \id_X)$$B \times X := (B_r \times X, \gamma_r \times X, g_r \times \id_X)$. The following simple lemma is clear but often useful.

Lemma 3.2. For any space $X$$X$ there is an isomorphism $\Omega_n^B(X) \cong \Omega_n^{B \times X}$$\Omega_n^B(X) \cong \Omega_n^{B \times X}$.

## 4 The orientation homomorphism

We fix a local orientation at the base-point of $BO$$BO$. It then follows that every closed $B$$B$-manifold $(M, \bar \nu)$$(M, \bar \nu)$ is given a local orientation. This amounts to a choice of fundamental class of $M$$M$ which is a generator

$\displaystyle [M] \in H_n(M; \underline{\Zz})$

where $\underline{\Zz}$$\underline{\Zz}$ denotes the local coefficient system defined by the orientation character of $M$$M$.

Given a closed $B$$B$-manifold $(M, \bar \nu)$$(M, \bar \nu)$ we can use $\bar \nu$$\bar \nu$ to push the fundamental class of $[M]$$[M]$ to $\bar \nu_*[M] \in H_n(B; \underline{\Zz})$$\bar \nu_*[M] \in H_n(B; \underline{\Zz})$. Now the local coefficient system is defined by the orientation character of the stable bundle $B$$B$. It is easy to check that $\bar \nu_*[M]$$\bar \nu_*[M]$ depends only on the $B$$B$-bordism class of $(M, \bar \nu)$$(M, \bar \nu)$ and is additive with respect to the operations $+/-$$+/-$ on $\Omega_n^B$$\Omega_n^B$.

Definition 4.1. Let $B$$B$ be a fibred stable vector bundle. The orientation homomorphism is defined as follows:

$\displaystyle \rho : \Omega_n^B \to H_n(B; \underline{\Zz}), ~~~[M, \bar \nu] \mapsto \bar \nu_*[M].$

For the singular bordism groups $\Omega_n^B(X)$$\Omega_n^B(X)$ we have no bundle over $X$$X$ so in general there is only a $\Zz/2$$\Zz/2$-valued orientation homomorphism. However, if the first Stiefel-Whitney class of $B$$B$ vanishes, $w_1(B) = 0$$w_1(B) = 0$, then all $B$$B$-manifolds are oriented in the usual sense and the orientation homomorphism can be lifted to $\Zz$$\Zz$.

Definition 4.2. Let $B$$B$ be a fibred stable vector bundle. The orientation homomorphism in singular bordism is defined as follows:

$\displaystyle \rho : \Omega_n^B(X) \to H_n(X; \Zz/2), ~~~ [(M, \bar \nu), f] \mapsto f_*[M].$

If $w_1(B) = 0$$w_1(B) = 0$ then for all closed $B$$B$-manifolds $[M] \in H_n(M; \Zz)$$[M] \in H_n(M; \Zz)$ and we can replace the $\Zz/2$$\Zz/2$-coefficients with $\Zz$$\Zz$-coefficients above.

## 5 The Pontrjagin-Thom isomorphism

If $E$$E$ is a vector bundle, let $T(E)$$T(E)$ denote its Thom space. Recall that that a fibred stable vector bundle $B = (B_r, \gamma_r, g_r)$$B = (B_r, \gamma_r, g_r)$ defines a stable vector bundle $(E_r, \gamma_r, \bar g_r)$$(E_r, \gamma_r, \bar g_r)$ where $E_r = \gamma_r^*(EO(r))$$E_r = \gamma_r^*(EO(r))$. This stable vector bundle defines a Thom spectrum which we denote $MB$$MB$. The $r$$r$-th space of $MB$$MB$ is $T(E_r)$$T(E_r)$.

By definition a $B$$B$-manifold, $(M, \bar \nu)$$(M, \bar \nu)$, is an equivalence class of $B_r$$B_r$-structures on $\nu(i)$$\nu(i)$, the normal bundle of an embedding $i : M \to \Rr^{n+r}$$i : M \to \Rr^{n+r}$. Hence $(M, \bar \nu)$$(M, \bar \nu)$ gives rise to the collapse map

$\displaystyle c(M, \bar \nu) : S^{n+r} \to T(E_r)$

where we identify $S^{n+r}$$S^{n+r}$ with the one-point compatificiation of $\Rr^{n+r}$$\Rr^{n+r}$, we map via $\bar \nu_r$$\bar \nu_r$ on a tubular neighbourhood of $i(M) \subset \Rr^{n+r}$$i(M) \subset \Rr^{n+r}$ and we map all other points to the base-point of $T(E_r)$$T(E_r)$. As r increases these maps are compatibly related by suspension and the structure maps of the spectrum $MB$$MB$. Hence we obtain a homotopy class

$\displaystyle [c(M, \bar \nu)] =: P((M, \bar \nu)) \in \text{lim}_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$

The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$$P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$$(M, \bar \nu)$.

Theorem 5.1. There is an isomorphism of abelian groups

$\displaystyle P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto P([M, \bar \nu]).$

For the proof see [Bröcker&tom Dieck1970, Satz 3.1 and Satz 4.9].

For example, if $B = PBO$$B = PBO$ is the path fibration over $BO$$BO$, then $MB$$MB$ is homotopic to the sphere spectrum $S$$S$ and $\pi_n(S) = \pi_n^S$$\pi_n(S) = \pi_n^S$ is the $n$$n$-th stable homotopy group. On the other hand, in this case $\Omega_n^B = \Omega_n^{fr}$$\Omega_n^B = \Omega_n^{fr}$ is the framed bordism group and as a special case of Theorem 5.1 we have

Theorem 5.2. There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$$P : \Omega_n^{fr} \cong \pi_n^S$.

The Pontrjagin-Thom isomorphism generalises to singular bordism.

Theorem 5.3. For any space $X$$X$ there is an isomorphism of abelian groups

$\displaystyle P : \Omega_n^B(X) \cong \pi_n^S(MB \wedge X_+)$

where $MB \wedge X_+$$MB \wedge X_+$ denotes the smash produce of the specturm $MB$$MB$ and the space $X$$X$ with a disjoint basepoint added.

## 6 Spectral sequences

For any generalised homology theory $h_*$$h_*$ there is a spectral sequence, called the Atiyah-Hirzebruch spectral sequence (AHSS) which can be used to compute $h_*(X)$$h_*(X)$. The $E_2$$E_2$ term of the AHSS is $H_p(X; h_q(\text{pt}))$$H_p(X; h_q(\text{pt}))$ and one writes

$\displaystyle \bigoplus_{p+q = n} H_p(X; h_q(\text{pt})) \Longrightarrow h_{n}(X).$

The Pontrjagin-Thom isomorphisms above therefore give rise to the following theorems. For the first we recall that stable homotopy defines a generalised homology theory, and we use the Thom isomorphism with local coefficients: $H_*(MB;A)\cong H_*(B;A_\omega)$$H_*(MB;A)\cong H_*(B;A_\omega)$.

Theorem 6.1. Let $B$$B$ be a fibred stable vector bundle. There is a spectral sequence

$\displaystyle \bigoplus_{p+q = n} H_p(B;\underline{\pi_q^S}) \Longrightarrow \Omega_{n}^B.$

Theorem 6.2. Let $B$$B$ be a fibred stable vector bundle and $X$$X$ a space. There is a spectral sequence

$\displaystyle \bigoplus_{p+q = n} H_p(X; \Omega_q^B) \Longrightarrow \Omega_n^B(X).$

Next recall Serre's theorem [Serre1951] that $\pi_i^S \otimes \Qq$$\pi_i^S \otimes \Qq$ vanishes unless $i=0$$i=0$ in which case $\pi_0^S \otimes \Qq \cong \Qq$$\pi_0^S \otimes \Qq \cong \Qq$. From the above spectral sequences of Theorems 6.1 and 6.2 we deduce the following

Theorem 6.3 Cf. [Kreck&Lück2005, Thm 2.1]. If $w_1(B) = 0$$w_1(B) = 0$ then the orientation homomorphism induces an isomorphism

$\displaystyle \rho \otimes \id_{\Qq} : \Omega_n^B \otimes \Qq \cong H_n(B; \Qq).$

Moreover for any space $X$$X$, $\Omega_n^B(X) \otimes \Qq \cong \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq))$$\Omega_n^B(X) \otimes \Qq \cong \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq))$ and if $B$$B$ is connected, the rationalised orientation homomorphism $\rho \otimes \id_{\Qq} : \Omega_n^B(X) \otimes \Qq \to H_n(X; \Qq)$$\rho \otimes \id_{\Qq} : \Omega_n^B(X) \otimes \Qq \to H_n(X; \Qq)$ may be identified with the projection

$\displaystyle \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq)) \to H_n(B; H_0(B; \Qq)) = H_n(B; \Qq).$

## 7 Piecewise linear and topological bordism

Let $BPL$$BPL$ and $BTOP$$BTOP$ denote respectively the classifying spaces for stable piecewise linear homeomorphisms of Euclidean space and origin-preserving homeomorphisms of Euclidean space. Note that while there are honest groups $TOP(n) = \text{Homeo}(\Rr^n, *)$$TOP(n) = \text{Homeo}(\Rr^n, *)$ and $TOP = \text{lim}_{n \to\infty} TOP(n)$$TOP = \text{lim}_{n \to\infty} TOP(n)$, the piecewise linear case requires more care.

If $CAT = PL$$CAT = PL$ or $TOP$$TOP$, and $\gamma : B \to BCAT$$\gamma : B \to BCAT$ is a fibration, and $M$$M$ is a compact $CAT$$CAT$ manifold then just as above, we can define an $B$$B$-structure on $M$$M$ to be an equivalence class of lifts of of the classifying map of the stable normal bundle of $M$$M$:

$\displaystyle \xymatrix{ & B \ar[d]^{\gamma} \\ M \ar[r]^{\nu_M} \ar[ur]^{\bar \nu} & BCAT.}$

Note that $CAT$$CAT$ manifolds have stable normal $CAT$$CAT$ bundles classified by $\nu_M \to BCAT$$\nu_M \to BCAT$.

Just as before we obtain bordism groups $\Omega_n^B$$\Omega_n^B$ of closed n-dimensional $CAT$$CAT$-manifolds with $B$$B$ structure

$\displaystyle \Omega_n^B : = \{ [M, \bar \nu ]\}.$

The fibration $B$$B$ again defines a Thom spectrum $MB$$MB$ and one asks if there is a Pontrjagin-Thom isomorphism. The proof of the Pontrjagin-Thom theorem relies on transversality for manifolds and while this is comparatively easy in the $PL$$PL$-category, it is was a major breakthrough to achieve this for topological manifolds: achieved in [Kirby&Siebenmann1977] for dimensions other than 4 and then in [Freedman&Quinn1990] in dimension 4. Thus one has

Theorem 7.1. There is an isomorphism $\Omega_n^B \cong \pi_n^S(MB)$$\Omega_n^B \cong \pi_n^S(MB)$.

The basic bordism groups for $PL$$PL$ and $TOP$$TOP$ manifolds, $B = (BCAT = BCAT)$$B = (BCAT = BCAT)$ and $B = (BSCAT \to BCAT)$$B = (BSCAT \to BCAT)$, are denoted by $\Omega_*^{PL}$$\Omega_*^{PL}$, $\Omega_*^{SPL}$$\Omega_*^{SPL}$, $\Omega_*^{TOP}$$\Omega_*^{TOP}$ and $\Omega_*^{STOP}$$\Omega_*^{STOP}$. Their computation is significantly more difficult than the corresponding bordism groups of smooth manifolds: there is no analogue of Bott periodicity for $\pi_i(PL)$$\pi_i(PL)$ and $\pi_i(TOP)$$\pi_i(TOP)$ and so the spectra $MPL$$MPL$ and $MTOP$$MTOP$ are far more complicated. For now we simply refer the reader to [Madsen&Milgram1979, Chapters 5 & 14] and [Brumfiel&Madsen&Milgram1973].

However, working rationally, the natural maps $O \to PL$$O \to PL$ and $O \to TOP$$O \to TOP$ induce isomorphisms
$\displaystyle \pi_i(O) \otimes \Qq \cong \pi_i(PL) \otimes \Qq ~~ \text{and} ~~ \pi_i(O) \otimes \Qq \cong \pi_i(TOP) \otimes \Qq ~~\forall i.$

As a consequence one has

Theorem 7.2. There are isomorphisms

$\displaystyle \Omega_i^{SO} \otimes \Qq \cong \Omega_i^{SPL} \otimes \Qq \cong \Omega_i^{STOP} \otimes \Qq ~~ \forall i.$