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
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 .
3 Stable vector bundles
We recall the defintion of a stable vector bundle from [Kreck&Lück2005, 18.10] which requires some notation. Let denote the trivial rank k bundle : the space will be clear from context. Let be a sequence of inclusions of CW-complexes. A stable vector bundle over is a sequence of rank k-vector bundles and a sequence of bundle maps
where covers the inclusion . An isomorphism of stable vector bundles over , is a sequence of bundle ismorphisms for , some integer, which are compatible with and .
Here are some important examples:
- A rank -vector over a CW complex defines a stable vector bundle by setting fixed for all , for and for with the obvious inclusions.
- If is a compact n-manifold then embedds into for and for two such embeddings are isotopic. The normal bundle of any such embedding defines a stable vector bundle over as in the first example and there is a canonical isomorphicm between any two such stable bundles given by isotopies between the embeddings. We write for the stable normal bundle of . Note that by defintion is a stable inverse to the tangent bundle of :
- The universal bundle is the stable bundle with , the universal -plane bundle and the classifymap of .
- If is a stable vector bundle over and is a sequence of maps compatible with inclusions then is a stable vector bundle over called the pull-back bundle and denoted .
4 B-manifolds and B-bordism
Let be a stable vector bundle. A normal -manifold is a triple consisting of a compact manifold , a sequence of maps , and a stable bundle isomorphism .
Recall that is a fibration. We next define normal -manifolds and normal -bordism: more details are given in [Stong1968][Chapter II] and [Kreck1999][Section 1]. If is a compact smooth manifold then is classified by a map, also denoted , : . A -structure on is a map lifting through :
A -manifold is a pair where is a -structure on . If has boundary , a disjoint union of two closed -manifolds with inclusions , , then is a -bordism between the -manifolds and .
5 The Pontrjagin Thom isomorphism
6 Spectral sequences
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