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 a deatiled treatment in [Stong1968, Chapter II] and a summary in [Kreck1999, Section 1] as well as [Kreck&Lück2005, 18.10].
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 fibraion . A -manifold is a compact manifold together with a lift of a classifying map for the stable normal bundle of to :
The n-dimensional -bordism group is defined to be the set of closed -manifolds up modulo the relation of -bordism and addition given by disjoint union
Alternative notations are and also when for a stable represenation of a topological group . Details of the definition and some important theorems for computing follow.
2 B-structures
In this section we give a compressed accont of [Stong1968, Chapter II]. Let denote the Grassman manifold of unoriented r-planes in and let be the infinite Grassman 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: an specific isomorphism, up to appropriate equivalence, is required to give a map between the set of structures. Happily this is the case for the normal bundle 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. 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 structures of the normal bundles of any two embeddings .
Now let be a sequence of fibrations over with maps fitting into the following commutative diagram
where is the standard inclusion and let . A -structure on the normal bundle of an embedding defines a unique -structure on the composition of with the standard inclusion .
Defition 2.3. 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 such that restricting to on . The restriction of this -structure to is denoted : by construction is the boundary of .
Definition 2.4. Closed -manifolds and are -bordant if there is a compact -manifold such that . We write for the bordism class of .
Proposition 2.5. The set of -borism class of closed n-manifolds with -structure,
forms an abelian group under the operation of disjoint union with inverse .
3 The Pontrjagin Thom isomorphism
If is a vector bundle, let denote its Thom space. Recall that is a sequence of fibrations with compatible maps . These data give rise to a stable vector bundle as defined in [Kreck&Lück2005, 18.10] with equal to the pullback bundle where . This stable vector bundle defines a Thom spectrum which we denote . The r-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 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 increase 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 .
Theorem 3.1. There is an isomorphism of abelian groups
4 Spectral sequences
5 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