B-Bordism

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1 Introduction

On this page we recall the definition of the bordism groups of closed manifolds, with extra topological structure: orientation, spin-structure, weak complex structure etc. The ideas here date back to [Lashof1965]. There is are detailed treatment in [Stong1968, Chapter II] and summaries in [Kreck&Lück2005, 18.10] and [Kreck1999, Section 1]. See also the Wikipedia bordism page.

We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$${{Stub}} == Introduction == <wikitex>; On this page we recall the definition of the bordism groups of closed manifolds, with extra topological structure: orientation, spin-structure, weak complex structure etc. The ideas here date back to {{cite|Lashof1965}}. There is are detailed treatment in {{cite|Stong1968|Chapter II}} and summaries in {{cite|Kreck&Lück2005|18.10}} and {{cite|Kreck1999|Section 1}}. See also the Wikipedia [[Wikipedia:Bordism|bordism page]]. We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$ where $BO$ denotes the classifying space of the stable orthogonal group and $B$ is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes $B$ for the fibration $\gamma$. Speaking somewhat imprecisely (precise details are below) a $B$-manifold is a compact manifold $M$ together with a lift of a classifying map for the stable normal bundle of $M$ to $B$: $$ \xymatrix{ & B \ar[d]^{\gamma} \ W \ar[r]_{\nu_W} \ar[ur]^{\bar \nu} & BO.} $$ The n-dimensional $B$-bordism group is defined to be the set of closed $B$-manifolds up modulo the relation of bordism via compact $B$-manifolds. Addition given by disjoint union and in fact for each $n >0$ there is a group $$ \Omega_n^B := \{ (M, \bar \nu) \}/\simeq.$$ Alternative notations are $\Omega_n(B)$ and also $\Omega_n^G$ when $B \to BO = BG \to BO$ for $G \to O$ a stable represenation of a topological group $G$. Details of the definition and some important theorems for computing $\Omega_n^B$ follow. </wikitex> === Examples === <wikitex>; We list some fundamental examples with common notation and also the fibration B. * [[Unoriented bordism|Unoriented bordism]]: $\mathcal{N}_*$; $B = (BO = BO)$. * [[Oriented bordism|Oriented bordism]]: $\Omega_*$, $\Omega_*^{SO}$; $B = (BSO \to BO)$. * [[Spin bordism|Spin bordism]]: $\Omega_*^{Spin}$; $B = (BSpin \to BO)$. * [[String bordism|String bodism]] : $\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$; $B = (BO\langle 8 \rangle \to BO)$. * [[Unitary bordism|Unitary bordism]] : $\Omega_*^U$; $B = (BU \to BO)$. * [[Special unitary bordism|Special unitary bordism]] : $\Omega_*^{SU}$; $B = (BSU \to BO)$. </wikitex> == B-structures and bordisms == <wikitex>; In this section we give a compressed accont of parts of {{cite|Stong1968|Chapter II}}. Let $G_{r, m}$ denote the [[Wikipedia:Grassman_manifold|Grassman manifold]] of unoriented r-planes in $\Rr^n$ and let $BO(r) = lim_{m \to \infty} G_{r, m}$ be the infinite Grassman and fix a fibration $\gamma_r : B_r \to BO(r)$. {{beginthm|Definition}} Let $\xi: E \to X$ be a rank r vector bundle classified by $\xi : X \to BO(r)$. A $B_r$-structure on $\xi$ is a vertical homotopy class of maps $\bar \xi : X \to B_r$ such that $\gamma_r \circ \bar \xi = \xi$. {{endthm}} Note that if $\xi_0$ and $\xi_1$ are isomorphic vector bundles over $X$ then the sets of $B_r$-structures on each are in bijective equivalence. However $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 set of $B_r$ structures. Happily this is the case for the normal bundle of an embedding as we now explain.| Let $M$ be a compact manifold and let $i_0 : M \to \Rr^{n+r}$ be an embedding. Equipping $\Rr^{n+r}$ with the standard metric, the [[Wikipedia:Normal_bundle|normal bundle]] of $i_0$ is a rank r vector bundle over classified by its normal Gauss map $\nu(i_0) : M \to G_{r, n+r} \subset BO(r)$. If $i_1$ is another such embedding and $r >> n$, then $i_1$ is [[Wikipedia:Regular_homotopy|regularly homotopic]] to $i_0$ and all regular homotopies are regularly homotopic relative to their endpoints. A regular homotopy $H$ defines an isomorphism $\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 {{beginthm|Lemma|{{cite|Stong1968|p 15}}}} For r sufficiently large, (depending only on n) there is a 1-1 correspondence between the $B_r$ structures of the normal bundles of any two embeddings $i_0, i_1 : M \to \Rr^{n+r}$. {{endthm}} This lemma is one motivataion for the useful, but subtle, notion of a fibred stable vector bundle. {{beginthm|Definition}} A fibred stable vector bundle $B$ consists of the following data: sequence of fibrations over $BO(r)$, $(B_r, \gamma_r)$ and a sequence of maps $g_r : B_r \to B_{r+1}$ fitting into the following commutative diagram $$ \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$ is the standard inclusion and let $B = lim_{r \to \infty}(B_r)$. {{endthm}} {{beginthm|Remark}} A fibred stable vector bundle $B$ gives rise to a stable vector bundle as defined in {{cite|Kreck&Lück2005|18.10}}. One defines $E_r \to B_r$ to be the pullback bundle $\gamma_r^*(EO(r))$ where $EO(r)$ is the universal r-plane bundle over $BO(r)$. Then the maps $g_r$ define bundle maps $\bar g_r : E_r \times \Rr \to E_{r+1}$ covering the maps $g_r$. {{endthm}} Now a $B_r$-structure on the normal bundle of an embedding $i: M \to \Rr^{n+r}$ defines a unique $B_{r+1}$-structure on the composition of $i$ with the standard inclusion $\Rr^{n+r} \to \Rr^{n+r+1}$. Hence we can make the following {{beginthm|Definition|{{cite|Stong1968|p 15}}}} Let $B$ be a fibred stable vectore bundle. A $B$-structure on $M$ is an equivalence class of $B_r$-structure on $M$ where two such structures are equivalent if they become equivalent for r sufficiently large. A $B$-manifold is a pair $(M, \bar \nu)$ where $M$ is a compact manifold and $\bar \nu$ is a $B$-structure on $M$. {{endthm}} If $W$ is a compact manifold with boundary $\partial W$ then by choosing the inward-pointing normal vector along $\partial W$, a $B$-structure on $W$ restricts to a $B$-structure on $\partial W$. In particular, if $(M, \bar \nu_M)$ is a closed $B$ manifold then $W = M \times [0, 1]$ has a canonical $B$-structure $\bar \nu_{M \times [0, 1]}$ such that restricting to $(M, \bar \nu_M)$ on $M \times \{ 0 \}$. The restriction of this $B$-structure to $M \times \{ 1 \}$ is denoted $-\bar \nu$: by construction $(M \sqcup M, \bar \nu \sqcup - \bar \nu)$ is the boundary of $(M \times [0, 1], \bar \nu_{M \times [0, 1]})$. {{beginthm|Definition}} Closed $B$-manifolds $(M_0, \bar \nu_0)$ and $(M_1, \bar \nu_1)$ are $B$-bordant if there is a compact $B$-manifold $(W, \bar \nu)$ such that $\partial(W, \bar \nu) = (M_0 \sqcup M_1, \bar \nu_0 \sqcup -\bar \nu_1)$. We write $[M, \bar \nu]$ for the bordism class of $(M, \bar \nu)$. {{endthm}} {{beginthm|Proposition|{{cite|Stong1968|p 17}}}} The set of $B$-borism class of closed n-manifolds with $B$-structure, $$ \Omega_n^B := \{ [M, \bar \nu ] \},$$ forms an abelian group under the operation of disjoint union with inverse $-[M,\bar \nu] = [M, -\bar \nu]$. {{endthm}} </wikitex> == Singular bordism == <wikitex>; $B$-bordism gives rise to a generalised homology theory. If $X$ is a space then the n-cycles of this homology theory are pairs $$ ((M, \bar \nu),~ f: M \to X) $$ where $(M, \bar \nu)$ is a closed n-dimensional $B$-manifold and $f$ is any continuous map. Two cycles $((M_0, \bar \nu_0), f_0)$ and $((M_1, \bar \nu_1), f_1)$ are homologous if there is a pair $$ ((W, \bar \nu),~ g : W \to X)$$ where $(W, \bar \nu)$ is a $B$-bordism from $(M_0, \bar \nu_0)$ to $(M_1, \bar \nu_1)$ and $g : W \to X$ is a continuous map extending $f_0 \sqcup f_1$. Writing $[(M, \bar \nu), f]$ for the equivalence class of $((M, \bar \nu) ,f)$ we obtain an abelian group $$ \Omega_n^B(X) : = \{ [(M, \bar \nu), f] \} $$ with group operation disjoint union and inverse $-[(M, \bar \nu), f] = [(M, - \bar \nu), f]$. {{beginthm|Proposition}} The mapping $X \to \Omega_n^B(X)$ defines a generalised homology theory with coefficients $\Omega_n^B(pt) = \Omega_n^B$. {{endthm}} Given a stable vector bundle $B = B_r, \gamma_r, g_r)$ we can form the stable vectore bundle $B \times X := (B_r \times X, \gamma_r \times X, g_r \times \id_X)$. The following simple lemma is clear but often useful. {{beginthm|Lemma}} For any space $X$ there is an isomorphism $\Omega_n^B(X) \cong \Omega_n^{B \times X}$. {{endthm}} </wikitex> == The Hurewicz homomorphism == <wikitex>; We fix a local orientation at the base-point of $BO$. It then follows that every closed $B$-manifold $(M, \bar \nu)$ is given a local orientation which amounts to a choice of fundamental class of $M$ which is a genator $$[M] \in H_n(M; \Zz_\omega)$$ where $\Zz_\omega$ denotes the local coefficient system defined by the orientation character of $M$. Given a closed $B$-manifold $(M, \bar \nu)$ we can use $\bar \nu$ to push the fundamental class of $[M]$ to $\bar \nu_*[M] \in H_n(B; \Zz_\omega)$: now the local coefficient system is defined by the orientation character of the stable bundle $B$. It is easy to check that $f_*[M]$ denepnds only on the $B$-bordism class of $(M, \bar \nu)$ and is additive with respect to the operations $+$ and $-$ on $\Omega_n^B$. {{beginthm|Definition}} Let $B$ be a fibred stable vector bundle. The Hurewicz homomorphism is defined as follows: $$ \rho : \Omega_n^B \to H_n(B; \Zz_\omega), ~~~[M, \bar \nu] \mapsto \bar \nu_*[M].$$ {{endthm}} For the singular bordism groups $\Omega_n^B(X)$ we have no bundle over $X$ so in general there is only a $\Zz/2$-valued Hurewicz homomophism. However, if $w_1(B) = 0$, then all $B$-manifolds are oriented in the usual sense and the Hurewicz homomorphism can be lifted to $\Zz$. {{beginthm|Definition}} Let $B$ be a fibred stable vector bundle. The Hurewicz homomorphism in singular bordism is defined as follows: $$ \rho : \Omega_n^B(X) \to H_n(X; \Zz/2), ~~~ [(M, \bar \nu), f] \mapsto f_*[M].$$ If $w_1(B) = 0$ then for all close $B$-manifolds $[M] \in H_n(M; \Zz)$ and we can replace the $\Zz/2$-coefficients with $\Zz$-coefficients above. {{endthm}} </wikitex> == The Pontrjagin-Thom isomorphism == <wikitex>; If $E$ is a vector bundle, let $T(E)$ denote its [[Wikipedia:Thom_space|Thom space]]. Recall that that a fibred stable vector bundle $B = (B_r, \gamma_r, g_r)$ defines a stable vector bundle $(E_r, \gamma_r, \bar g_r)$ where $E_r = \gamma_r^*(EO(r))$. This stable vector bundle defines a Thom [[Wikipedia:Spectrum(homotopy_theory)|spectrum]] which we denote $MB$. The r-th space of $MB$ is $T(E_r)$. By definition a $B$-manifold, $(M, \bar \nu)$, is an equivalence class of $B_r$-structures on $\nu(i)$, the normal bundle of an embedding $i : M \to \Rr^{n+r}$. Hence $(M, \bar \nu)$ gives rise to the collapse map $$ c(M, \bar \nu) : S^{n+r} \to T(E_r)$$ where we identify $S^{n+r}$ with the [[Wikipedia:One-point_compactification|one-point compatificiation]] of $\Rr^{n+r}$, we map via $\bar \nu_r$ on a tubular neighbourhood of $i(M) \subset \Rr^{n+r}$ and we map all other points to the base-point of $T(E_r)$. As r increases these maps are compatibly related by suspension and the structure maps of the spectrum $MB$. Hence we obtain a homotopy class $$ [c(M, \bar \nu)] =: P((M, \bar \nu)) \in lim_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$$ The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$. {{beginthm|Theorem}} There is an isomorphism of abelian groups $$ P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto P([M, \bar \nu]).$$ {{endthm}} For example, if $B_r = pt$ is the one-point space for each r, then $MB$ is the sphere spectrum $S$ and $\pi_n(S) = \pi_n^S$ is the [[Wikipedia:Stable_homotopy_groups_of_spheres|n-th stable homotopy group]]. On the other hand, in this case $\Omega_n^B = \Omega_n^{fr}$ is the framed bordism group and as special case we have {{beginthm|Theorem}} There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$. {{endthm}} The Pontrjagin-Thom isomorphism generalises to singular bordism. {{beginthm|Theorem}} For any space $X$ there is an isomorphism of abelian groups $$ P : \Omega_n^B(X) \cong \pi_n^S(MB \wedge X)$$ where $MB \wedge X$ denotes the smash produce of the specturm $MB$ and the space $X$. {{endthm}} </wikitex> == Spectral sequences == <wikitex>; For any generalised homology thoery there $h_*$ there is a spectral sequence, called the Atiyah-Hirzeburbh spectral sequence (AHSS) which can be used to compute. $h_*(X)$. The $E_2$ term of the AHSS is $H_p(X; h_q(pt))$ and one writes $$ \bigoplus_{p+q = n} H_p(B; h_q(pt)) \Longrightarrow h_{n}(X).$$ The Pontryagin-Thom isomorphisms above therefore give rise to the following theorems. For the first we recall that stable homotopy defines a generalised homology theory. {{beginthm|Theorem}} Let $B$ be a fibred stable vector bundle. There is a spectral sequence $$ \bigoplus_{p+q = n} H_p(B; \pi_q^S) \Longrightarrow \Omega_{n}^B$$ {{beginthm|Theorem}} Let $B$ be a fibred stable vector bundle and $X$ a space. There is a spectral sequence $$ \bigoplus_{p+q = n} H_p(B; \Omega_q^B) \Longrightarrow \Omega_n^B(X).$$ {{endthm}} Next recall Serre's theorem $\pi_i^S \otimes \Qq$ vanishes unless $i=0$ in which case $\pi_0^S \otimes \Qq \cong \Qq$. From the above spectral sequences above we deduce the following {{beginthm|Theorem|{{cite|Kreck&Lück2005|Thm 2.1}}}} If $w_1(B) = 0$ then the Hurewicz homomorphism induces an isomorphism $$ \rho \otimes \id_{\Qq} : \Omega_n^B \cong H_n(B; \Qq).$$ Moreover for any space $X$, $\Omega_n^B(X) \cong \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq))$ and if $B$ is connected, the rationaliesed Hurewicz homorphism $\rho \otimes \id_{\Qq} : \Omega_n^B(X) \otimes \Qq \to H_n(X; \Qq)$ maybe identified with the projection $$ \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq)) \to H_n(B; H_0(B; \Qq) = H_n(B; \Qq).$$ {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Theory]]\gamma : B \to BO$ where $BO$$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$$M$ together with a lift of a classifying map for the stable normal bundle of $M$$M$ to $B$$B$:

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

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

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

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 represenation 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 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)$$B = (BSpin \to BO)$.
• String bodism : $\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$$\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$; $B = B\langle 8 \rangle to BO$$B = B\langle 8 \rangle to BO$.
• Unitary bordism : $\Omega_*^U$$\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$.

2 B-structures and bordisms

In this section we give a compressed accont of parts of [Stong1968, Chapter II]. Let $G_{r, m}$$G_{r, m}$ denote the Grassman manifold of unoriented r-planes in $\Rr^n$$\Rr^n$ and let $BO(r) = lim_{m \to \infty} G_{r, m}$$BO(r) = lim_{m \to \infty} G_{r, m}$ be the infinite Grassman and fix a fibration $\gamma_r : B_r \to BO(r)$$\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$$B_r$-structures are defined on specific bundles, not isomorphism classes of bundles: a specific isomorphism, up to appropriate equivalence, is required to give a bijection between the set of $B_r$$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 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$$i_0$ and all regular homotopies are regularly homotopic relative to their endpoints. A regular homotopy $H$$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 $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 motivataion for the useful, but subtle, notion of a fibred stable vector bundle.

Definition 2.3. A fibred stable vector bundle $B$$B$ consists of the following data: sequence of fibrations over $BO(r)$$BO(r)$, $(B_r, \gamma_r)$$(B_r, \gamma_r)$ and a sequence of maps $g_r : B_r \to B_{r+1}$$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$$j_r$ is the standard inclusion and let $B = lim_{r \to \infty}(B_r)$$B = 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)$$BO(r)$. Then the maps $g_r$$g_r$ define bundle maps $\bar g_r : E_r \times \Rr \to E_{r+1}$$\bar g_r : E_r \times \Rr \to E_{r+1}$ covering the maps $g_r$$g_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$$B$ be a fibred stable vectore bundle. A $B$$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]}$ such that restricting 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$$B$-borism class of closed n-manifolds with $B$$B$-structure,

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

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-cycles of this homology theory are pairs

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

where $(M, \bar \nu)$$(M, \bar \nu)$ is a closed n-dimensional $B$$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

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

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

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

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)$$X \to \Omega_n^B(X)$ defines a generalised homology theory with coefficients $\Omega_n^B(pt) = \Omega_n^B$$\Omega_n^B(pt) = \Omega_n^B$.

Given a stable vector bundle $B = B_r, \gamma_r, g_r)$$B = B_r, \gamma_r, g_r)$ we can form the stable vectore bundle $B \times X := (B_r \times X, \gamma_r \times X, g_r \times \id_X)$$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 Hurewicz 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)$$(M, \bar \nu)$ is given a local orientation which amounts to a choice of fundamental class of $M$$M$ which is a genator

$\displaystyle [M] \in H_n(M; \Zz_\omega)$ $$\displaystyle [M] \in H_n(M; \Zz_\omega)$$

where $\Zz_\omega$$\Zz_\omega$ denotes the local coefficient system defined by the orientation character of $M$$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; \Zz_\omega)$$\bar \nu_*[M] \in H_n(B; \Zz_\omega)$: now the local coefficient system is defined by the orientation character of the stable bundle $B$$B$. It is easy to check that $f_*[M]$$f_*[M]$ denepnds only on the $B$$B$-bordism class of $(M, \bar \nu)$$(M, \bar \nu)$ and is additive with respect to the operations $+$$+$ and $-$$-$ on $\Omega_n^B$$\Omega_n^B$.

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

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

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$$\Zz/2$-valued Hurewicz homomophism. However, if $w_1(B) = 0$$w_1(B) = 0$, then all $B$$B$-manifolds are oriented in the usual sense and the Hurewicz homomorphism can be lifted to $\Zz$$\Zz$.

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

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

If $w_1(B) = 0$$w_1(B) = 0$ then for all close $B$$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-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

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

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

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

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

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

For example, if $B_r = pt$$B_r = pt$ is the one-point space for each r, then $MB$$MB$ is the sphere spectrum $S$$S$ and $\pi_n(S) = \pi_n^S$$\pi_n(S) = \pi_n^S$ is the 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 special case 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

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

where $MB \wedge X$$MB \wedge X$ denotes the smash produce of the specturm $MB$$MB$ and the space $X$$X$.

6 Spectral sequences

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

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

The Pontryagin-Thom isomorphisms above therefore give rise to the following theorems. For the first we recall that stable homotopy defines a generalised homology theory.

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

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

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

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

Next recall Serre's theorem $\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 above we deduce the following

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

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

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

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

7 References

• [Kreck&Lück2005] M. Kreck and W. Lück, The Novikov conjecture, Birkhäuser Verlag, Basel, 2005. MR2117411 (2005i:19003) Zbl 1058.19001
• [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
• [Lashof1965] R. Lashof, Problems in differential and algebraic topology. Seattle Conference, 1963, Ann. of Math. (2) 81 (1965), 565–591. MR0182961 (32 #443) Zbl 0137.17601
• [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010