B-Bordism
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− | We briefly recall the defintion of $B$-manifolds and $B$-bordism: more details are given in {{cite|Stong1968}}[Chapter II] and {{cite|Kreck1999}}[Section 1]. If $M$ is a | + | We briefly recall the defintion of $B$-manifolds and $B$-bordism: more details are given in {{cite|Stong1968}}[Chapter II] and {{cite|Kreck1999}}[Section 1]. If $M$ is a compact smooth manifold then the stable normal bundle of $M$ is classified by a map $\nu_M : M \to BO$. A $B$-structure on $M$ is a map $\bar \nu: M \to B$ lifting $\nu_M$ through $B$: |
+ | $$ | ||
+ | \xymatrix{ | ||
+ | & B \ar[d]^{\gamma} \\ | ||
+ | M \ar[r]_{\nu_M}\ar[ur]^{\bar \nu} & BO} | ||
+ | $$ | ||
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Revision as of 21:12, 7 January 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Let denote the classifying space of the stable orthogonal group and let denote a fibration where is homotopy equivalent to a CW complex of finite type. For example, consider , or the maps or induced by the canonical homomorphisms and .
Associated to each fibration there is a bordism group of closed n-manifolds with structure denoted
The latter notation is used if is the classifying space of the group . In this page we review the the defintion bordism of the groups and recall some fundamental theorems about and properties of these groups.
2 B-manifolds and B-bordism
We briefly recall the defintion of -manifolds and -bordism: more details are given in [Stong1968][Chapter II] and [Kreck1999][Section 1]. If is a compact smooth manifold then the stable normal bundle of is classified by a map . A -structure on is a map lifting through :
3 The Pontrjagin Thom isomorphism
4 Spectral sequences
5 References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010