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[edit] 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 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:

\displaystyle  \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

\displaystyle  \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.

[edit] 1.1 Examples

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

[edit] 2 B-structures and bordisms

In this section we give a compressed accont of parts of [Stong1968, Chapter II]. Let G_{r, m} denote the 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).

Definition 2.1. 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.

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 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 regularly homotopic to i_0 and all regular homotopies are regularly homotopic relative to their endpoints (see [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

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 structures of the normal bundles of any two embeddings 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) 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

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

Remark 2.4. A fibred stable vector bundle B gives rise to a stable vector bundle as defined in [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.

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

Definition 2.5 [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.

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]}).

Definition 2.6. 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).

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

[edit] 3 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

\displaystyle  ((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

\displaystyle  ((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

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

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

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

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.

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

[edit] 4 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

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

where \underline{\Zz} 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; \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.

Definition 4.1. Let 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) we have no bundle over X so in general there is only a \Zz/2-valued orientation homomorphism. However, if the first 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.

Definition 4.2. Let 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 then for all closed B-manifolds [M] \in H_n(M; \Zz) and we can replace the \Zz/2-coefficients with \Zz-coefficients above.

[edit] 5 The Pontrjagin-Thom isomorphism

If E is a vector bundle, let T(E) denote its 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 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

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

where we identify S^{n+r} with the 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

\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)) depends only on the bordism class of (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 is the path fibration over BO, then MB is homotopic to the sphere spectrum S and \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} 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.

The Pontrjagin-Thom isomorphism generalises to singular bordism.

Theorem 5.3. For any space 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_+ denotes the smash produce of the specturm MB and the space X with a disjoint basepoint added.

[edit] 6 Spectral sequences

For any generalised homology theory h_* there is a spectral sequence, called the 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

\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).

Theorem 6.1. Let 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 be a fibred stable vector bundle and 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 vanishes unless i=0 in which case \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 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, \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

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

[edit] 7 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:

\displaystyle  \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

\displaystyle  \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 [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).

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 [Madsen&Milgram1979, Chapters 5 & 14] and [Brumfiel&Madsen&Milgram1973].

However, working rationally, the natural maps O \to PL and 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.

[edit] 8 References

[edit] 9 External links

  • The Encyclopedia of Mathematics article on bordism.
  • The Wikipedia page on cobordism.
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