# B-Bordism

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## 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$${{Stub}} == 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 [[Wikipedia:Piecewise_linear_manifold|piecewise linear]] and [[Wikipedia:Topological_manifold|topological manifolds]] is similar and we discuss it briefly [[B-Bordism#Piecewise linear and topological bordism|below]]. The formulation of the general set-up for B-Bordism dates back to {{cite|Lashof1963}}. There are detailed treatments in {{cite|Stong1968|Chapter II}} and {{cite|Bröcker&tom Dieck1970}} as well as summaries in {{cite|Teichner1992|Part 1: 1}}, {{cite|Kreck1999|Section 1}}, {{cite|Kreck&Lück2005|18.10}}. See also the [[Wikipedia:Bordism|Wikipedia 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 to B of a classifying map for the stable normal bundle of M: \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 modulo the relation of bordism via compact B-manifolds. Addition is given by disjoint union and in fact for each n \geq 0 there is a group \Omega_n^B := \{ (M, \bar \nu) \}/\equiv. Alternative notations are \Omega_n(B) and also \Omega_n^G when (B \to BO) = (BG \to BO) for G \to O a stable representation of a topological group G. Details of the definition and some important theorems for computing \Omega_n^B follow. === Examples === ; We list some fundamental examples with common notation and also indicate 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). * [[Spin^c bordism|Spinc bordism]]: \Omega_*^{Spin^{c}}; B = (BSpin^{c} \to BO). * [[String bordism|String bodism]] : \Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}; B = (BO\langle 8 \rangle \to BO). * [[Complex bordism|Complex bordism]] : \Omega_*^U; B = (BU \to BO). * [[Special unitary bordism|Special unitary bordism]] : \Omega_*^{SU}; B = (BSU \to BO). * [[Framed bordism|Framed bordism]] : \Omega_*^{fr}; B = (PBO \to BO), the path space fibration. == B-structures and bordisms == ; In this section we give a compressed accont of parts of {{cite|Stong1968|Chapter II}}. Let G_{r, m} denote the [[Wikipedia:Grassman_manifold|Grassmann manifold]] of unoriented r-planes in \Rr^m and let 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). {{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 sets 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 M 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 (see {{cite|Hirsch1959}}). 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 set of 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 motivation for the useful but subtle notion of a fibred stable vector bundle. {{beginthm|Definition}} A fibred stable vector bundle B = (B_r, \gamma_r, g_r) consists of the following data: a sequence of fibrations \gamma_r : B_r \to BO(r) together with 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. We let B = \text{lim}_{r \to \infty}(B_r). {{endthm}} {{beginrem|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). The diagram above gives rise to bundle maps \bar g_r : E_r \oplus \underline{\Rr} \to E_{r+1} covering the maps g_r; where \underline{\Rr} denotes the trivial rank 1 bundle over B_r. {{endrem}} 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 vector 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]} which restricts 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-bordism classes 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}} == Singular bordism == ; 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(\text{pt}) = \Omega_n^B. {{endthm}} Given a stable vector bundle 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). 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}} == The orientation homomorphism == ; 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. This amounts to a choice of fundamental class of M which is a generator [M] \in H_n(M; \underline{\Zz}) where \underline{\Zz} denotes the local coefficient system defined by the [[Wikipedia:Orientation_character|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; \underline{\Zz}). Now the local coefficient system is defined by the orientation character of the stable bundle B. It is easy to check that \bar \nu_*[M] depends only on the B-bordism class of (M, \bar \nu) and is additive with respect to the operations +/- on \Omega_n^B. {{beginthm|Definition}} Let B be a fibred stable vector bundle. The orientation homomorphism is defined as follows: \rho : \Omega_n^B \to H_n(B; \underline{\Zz}), ~~~[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 orientation homomorphism. However, if the first [[Wikipedia:Stiefel-Whitney_class|Stiefel-Whitney class]] of B vanishes, w_1(B) = 0, then all B-manifolds are oriented in the usual sense and the orientation homomorphism can be lifted to \Zz. {{beginthm|Definition}} Let B be a fibred stable vector bundle. The orientation 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 closed B-manifolds [M] \in H_n(M; \Zz) and we can replace the \Zz/2-coefficients with \Zz-coefficients above. {{endthm}} == The Pontrjagin-Thom isomorphism == ; 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 \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)) depends only on the bordism class of (M, \bar \nu). {{beginthm|Theorem}} \label{thm:PT-iso} 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 the proof see {{cite|Bröcker&tom Dieck1970|Satz 3.1 and Satz 4.9}}. For example, if B = PBO is the path fibration over BO, then MB is homotopic to 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 a special case of Theorem \ref{thm:PT-iso} 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 with a disjoint basepoint added. {{endthm}} == Spectral sequences == ; For any generalised homology theory h_* there is a spectral sequence, called the [[Wikipedia:Atiyah-Hirzebruch_spectral_sequence|Atiyah-Hirzebruch spectral sequence]] (AHSS) which can be used to compute h_*(X). The E_2 term of the AHSS is H_p(X; h_q(\text{pt})) and one writes \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). {{beginthm|Theorem}} Let B be a fibred stable vector bundle. There is a spectral sequence \bigoplus_{p+q = n} H_p(B;\underline{\pi_q^S}) \Longrightarrow \Omega_{n}^B. {{endthm}} {{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(X; \Omega_q^B) \Longrightarrow \Omega_n^B(X). {{endthm}} Next recall [[Wikipedia:Stable_homotopy_groups_of_spheres#Finiteness_and_torsion|Serre's theorem]] {{cite|Serre1951}} that \pi_i^S \otimes \Qq vanishes unless i=0 in which case \pi_0^S \otimes \Qq \cong \Qq. From the above spectral sequences of Theorems \ref{SS1} and \ref{SS2} we deduce the following {{beginthm|Theorem|Cf. {{cite|Kreck&Lück2005|Thm 2.1}}}} If w_1(B) = 0 then the orientation homomorphism induces an isomorphism \rho \otimes \id_{\Qq} : \Omega_n^B \otimes \Qq \cong H_n(B; \Qq). Moreover for any space X, \Omega_n^B(X) \otimes \Qq \cong \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq)) and if B is connected, the rationalised orientation homomorphism \rho \otimes \id_{\Qq} : \Omega_n^B(X) \otimes \Qq \to H_n(X; \Qq) may be 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}} == Piecewise linear and topological bordism == ; Let BPL and 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, *) and TOP = \text{lim}_{n \to\infty} TOP(n), the piecewise linear case requires more care. If CAT = PL or TOP, and \gamma : B \to BCAT is a fibration, and M is a compact CAT manifold then just as above, we can define an B-structure on M to be an equivalence class of lifts of of the classifying map of the stable normal bundle of M: \xymatrix{ & B \ar[d]^{\gamma} \ M \ar[r]^{\nu_M} \ar[ur]^{\bar \nu} & BCAT.} Note that CAT manifolds have stable normal CAT bundles classified by \nu_M \to BCAT. Just as before we obtain bordism groups \Omega_n^B of closed n-dimensional CAT-manifolds with B structure \Omega_n^B : = \{ [M, \bar \nu ]\}. The fibration B again defines a Thom spectrum 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-category, it is was a major breakthrough to achieve this for topological manifolds: achieved in {{cite|Kirby&Siebenmann1977}} for dimensions other than 4 and then in {{cite|Freedman&Quinn1990}} in dimension 4. Thus one has {{beginthm|Theorem}} There is an isomorphism \Omega_n^B \cong \pi_n^S(MB). {{endthm}} The basic bordism groups for PL and TOP manifolds, B = (BCAT = BCAT) and B = (BSCAT \to BCAT), are denoted by \Omega_*^{PL}, \Omega_*^{SPL}, \Omega_*^{TOP} and \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) and \pi_i(TOP) and so the spectra MPL and MTOP are far more complicated. For now we simply refer the reader to {{cite|Madsen&Milgram1979|Chapters 5 & 14}} and {{cite|Brumfiel&Madsen&Milgram1973}}. However, working rationally, the natural maps O \to PL and O \to TOP induce isomorphisms \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 {{beginthm|Theorem}} There are isomorphisms \Omega_i^{SO} \otimes \Qq \cong \Omega_i^{SPL} \otimes \Qq \cong \Omega_i^{STOP} \otimes \Qq ~~ \forall i. {{endthm}} == References == {{#RefList:}} == External links == * The Encyclopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Bordism bordism]. * The Wikipedia page on [[Wikipedia:Cobordism|cobordism]]. [[Category:Theory]] [[Category:Bordism]]\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

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