B-Bordism

Revision as of 15:11, 8 January 2010

1 Introduction

Let ${{Stub}} == Introduction == <wikitex>; Let $BO$ denote the classifying space of the stable orthogonal group and let $B$ denote a fibration $\gamma : B \to BO$ where $B$ is homotopy equivalent to a CW complex of finite type. For example, consider $\id : BO \to BO$, or the maps $BSO \to BO$ or $BU \to BO$ induced by the canonical homomorphisms $SO \to O$ and $U \to O$. Associated to each fibration $B$ there is a bordism group of closed n-manifolds with $B$ structure denoted $$\Omega_n^B ~~\text{or}~~ \Omega_n^G.$$ The latter notation is used if $B = BG$ is the classifying space of the group $G$. In this page we review the the defintion bordism of the groups $\Omega_n^B$ and recall some fundamental theorems about and properties of these groups. </wikitex> == B-manifolds and B-bordism == <wikitex> 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 $B$: $$ \xymatrix{ & B \ar[d]^{\gamma} \ M \ar[r]_{\nu_M}\ar[ur]^{\bar \nu} & BO} $$ </wikitex> == The Pontrjagin Thom isomorphism == <wikitex> $$ \Omega_n^B \cong \pi_n^S(MB)$$ </wikitex> == Spectral sequences == <wikitex> $$ H_p(B; \pi_q^S) \Longrightarrow \Omega_{p+q}^B$$ </wikitex> == References == {{#RefList:}} [[Category:Theory]]BO$ denote the classifying space of the stable orthogonal group and let $B$ denote a fibration $\gamma : B \to BO$ where $B$ is homotopy equivalent to a CW complex of finite type. For example, consider $\id : BO \to BO$, or the maps $BSO \to BO$ or $BU \to BO$ induced by the canonical homomorphisms $SO \to O$ and $U \to O$.

Associated to each fibration $B$ there is a bordism group of closed n-manifolds with $B$ structure denoted

$$\displaystyle \Omega_n^B ~~\text{or}~~ \Omega_n^G.$$

The latter notation is used if $B = BG$ is the classifying space of the group $G$. In this page we review the the defintion bordism of the groups $\Omega_n^B$ and recall some fundamental theorems about concerning these groups.

2 B-manifolds and B-bordism

We briefly recall the defintion of $B$-manifolds and $B$-bordism: more details are given in [Stong1968][Chapter II] and [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 $B$:

3 The Pontrjagin Thom isomorphism

$$\displaystyle \Omega_n^B \cong \pi_n^S(MB)$$

4 Spectral sequences

$$\displaystyle H_p(B; \pi_q^S) \Longrightarrow \Omega_{p+q}^B$$

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