B-Bordism
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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 where denotes the classifying space of the stable orthogonal group and is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes for the fibration . Speaking somewhat imprecisely (precise details are below) a -manifold is a compact manifold together with a lift to of a classifying map for the stable normal bundle of :
The -dimensional -bordism group is defined to be the set of closed -manifolds modulo the relation of bordism via compact -manifolds. Addition is given by disjoint union and in fact for each there is a group
Alternative notations are and also when for a stable representation of a topological group . Details of the definition and some important theorems for computing follow.
[edit] 1.1 Examples
We list some fundamental examples with common notation and also indicate the fibration B.
- Unoriented bordism: ; .
- Oriented bordism: , ; .
- Spin bordism: ; .
- Spin^{\Cc} bordism: ; .
- String bodism : ; .
- Complex bordism : ; .
- Special unitary bordism : ; .
- Framed bordism : ; , the path space fibration.
[edit] 2 B-structures and bordisms
In this section we give a compressed accont of parts of [Stong1968, Chapter II]. Let denote the Grassmann manifold of unoriented -planes in and let be the infinite Grassmannian and fix a fibration .
Definition 2.1. Let be a rank r vector bundle classified by . A -structure on is a vertical homotopy class of maps such that .
Note that if and are isomorphic vector bundles over then the sets of -structures on each are in bijective equivalence. However -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 structures. Happily this is the case for the normal bundle of an embedding as we now explain. Let be a compact manifold and let be an embedding. Equipping with the standard metric, the normal bundle of is a rank r vector bundle over classified by its normal Gauss map . If is another such embedding and , then is regularly homotopic to and all regular homotopies are regularly homotopic relative to their endpoints (see [Hirsch1959]). A regular homotopy defines an isomorphism 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 structures of the normal bundles of any two embeddings .
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 consists of the following data: a sequence of fibrations together with a sequence of maps fitting into the following commutative diagram
where is the standard inclusion. We let .
Remark 2.4. A fibred stable vector bundle gives rise to a stable vector bundle as defined in [Kreck&Lück2005, 18.10]. One defines to be the pullback bundle where is the universal r-plane bundle over . The diagram above gives rise to bundle maps covering the maps ; where denotes the trivial rank 1 bundle over .
Now a -structure on the normal bundle of an embedding defines a unique -structure on the composition of with the standard inclusion . Hence we can make the following
Definition 2.5 [Stong1968, p 15]. Let be a fibred stable vector bundle. A -structure on is an equivalence class of -structure on where two such structures are equivalent if they become equivalent for r sufficiently large. A -manifold is a pair where is a compact manifold and is a -structure on .
If is a compact manifold with boundary then by choosing the inward-pointing normal vector along , a -structure on restricts to a -structure on . In particular, if is a closed manifold then has a canonical -structure which restricts to on . The restriction of this -structure to is denoted : by construction is the boundary of .
Definition 2.6. Closed -manifolds and are -bordant if there is a compact -manifold such that . We write for the bordism class of .
Proposition 2.7 [Stong1968, p 17]. The set of -bordism classes of closed n-manifolds with -structure,
forms an abelian group under the operation of disjoint union with inverse .
[edit] 3 Singular bordism
-bordism gives rise to a generalised homology theory. If is a space then the -cycles of this homology theory are pairs
where is a closed -dimensional -manifold and is any continuous map. Two cycles and are homologous if there is a pair
where is a -bordism from to and is a continuous map extending . Writing for the equivalence class of we obtain an abelian group
with group operation disjoint union and inverse .
Proposition 3.1. The mapping defines a generalised homology theory with coefficients .
Given a stable vector bundle we can form the stable vector bundle . The following simple lemma is clear but often useful.
Lemma 3.2. For any space there is an isomorphism .
[edit] 4 The orientation homomorphism
We fix a local orientation at the base-point of . It then follows that every closed -manifold is given a local orientation. This amounts to a choice of fundamental class of which is a generator
where denotes the local coefficient system defined by the orientation character of .
Given a closed -manifold we can use to push the fundamental class of to . Now the local coefficient system is defined by the orientation character of the stable bundle . It is easy to check that depends only on the -bordism class of and is additive with respect to the operations on .
Definition 4.1. Let be a fibred stable vector bundle. The orientation homomorphism is defined as follows:
For the singular bordism groups we have no bundle over so in general there is only a -valued orientation homomorphism. However, if the first Stiefel-Whitney class of vanishes, , then all -manifolds are oriented in the usual sense and the orientation homomorphism can be lifted to .
Definition 4.2. Let be a fibred stable vector bundle. The orientation homomorphism in singular bordism is defined as follows:
If then for all closed -manifolds and we can replace the -coefficients with -coefficients above.
[edit] 5 The Pontrjagin-Thom isomorphism
If is a vector bundle, let denote its Thom space. Recall that that a fibred stable vector bundle defines a stable vector bundle where . This stable vector bundle defines a Thom spectrum which we denote . The -th space of is .
By definition a -manifold, , is an equivalence class of -structures on , the normal bundle of an embedding . Hence gives rise to the collapse map
where we identify with the one-point compatificiation of , we map via on a tubular neighbourhood of and we map all other points to the base-point of . As r increases these maps are compatibly related by suspension and the structure maps of the spectrum . Hence we obtain a homotopy class
The celebrated theorem of Pontrjagin and Thom states in part that depends only on the bordism class of .
Theorem 5.1. There is an isomorphism of abelian groups
For the proof see [Bröcker&tom Dieck1970, Satz 3.1 and Satz 4.9].
For example, if is the path fibration over , then is homotopic to the sphere spectrum and is the -th stable homotopy group. On the other hand, in this case is the framed bordism group and as a special case of Theorem 5.1 we have
Theorem 5.2. There is an isomorphism .
The Pontrjagin-Thom isomorphism generalises to singular bordism.
Theorem 5.3. For any space there is an isomorphism of abelian groups
where denotes the smash produce of the specturm and the space with a disjoint basepoint added.
[edit] 6 Spectral sequences
For any generalised homology theory there is a spectral sequence, called the Atiyah-Hirzebruch spectral sequence (AHSS) which can be used to compute . The term of the AHSS is and one writes
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: .
Theorem 6.1. Let be a fibred stable vector bundle. There is a spectral sequence
Theorem 6.2. Let be a fibred stable vector bundle and a space. There is a spectral sequence
Next recall Serre's theorem [Serre1951] that vanishes unless in which case . 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 then the orientation homomorphism induces an isomorphism
Moreover for any space , and if is connected, the rationalised orientation homomorphism may be identified with the projection
[edit] 7 Piecewise linear and topological bordism
Let and 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 and , the piecewise linear case requires more care.
If or , and is a fibration, and is a compact manifold then just as above, we can define an -structure on to be an equivalence class of lifts of of the classifying map of the stable normal bundle of :
Note that manifolds have stable normal bundles classified by .
Just as before we obtain bordism groups of closed n-dimensional -manifolds with structure
The fibration again defines a Thom spectrum 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 -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 .
The basic bordism groups for and manifolds, and , are denoted by , , and . Their computation is significantly more difficult than the corresponding bordism groups of smooth manifolds: there is no analogue of Bott periodicity for and and so the spectra and 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 and induce isomorphismsAs a consequence one has
Theorem 7.2. There are isomorphisms
[edit] 8 References
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