B-Bordism

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Revision as of 19:04, 18 January 2010

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1 Introduction

In this article we recall the definition of the bordism groups of closed manifolds, ususally with extra topological structure: orientation, spin-structure, weak complex structure etc.

We specify the extra structure by a fibration ${{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 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]]\xi : B \to BO$ where $BO$ denotes the classifying space of the stable orthogonal group and $B$ is homotopy equivalent to a CW complex of finite type. For example the fibrations $BSO \to BO$ and $BU \to BO$ induced by the canonical homomorphisms $SO \to O$ and $U \to O$ are used for respectively oriented and complex bordism.

Abusing notation, one writes $B$ for the fibraion $\xi$ : for each $B$ and $n \geq 0$ there is a bordism group of closed n-manifolds with $B$ structure denoted

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

If $B = BG$ is the classifying space of the stable group $G$ then on often writes

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

2 Stable vector bundles

We recall the defintion of a stable vector bundle from [Kreck&Lück2005, 18.10] which requires some notation. Let $\underline{\Rr^k}$ denote the trivial rank k bundle $Z \times \Rr^k$ : the space $Z$ will be clear from context. Let $\{ X_k \}$ be a sequence $X_0 \to X_1 \to X_2 \to \dots$ of inclusions $j_k : X_k \to X_{k+1}$ of CW-complexes. A stable vector bundle $E$ over $\{ X_k \}$ is a sequence of rank k-vector bundles $\xi_k \to X_k$ and a sequence of bundle maps

$\displaystyle \bar j_k : \xi_k \oplus \underline{\Rr} \to \xi_{k+1}$ $\displaystyle \bar j_k : \xi_k \oplus \underline{\Rr} \to \xi_{k+1}$

where $\bar j_k$ covers the inclusion $j_k$ . An isomorphism of stable vector bundles over $\{ X \}$ , $\theta : \{ (\xi_k\, \bar j_k) \} \cong \{\xi_k', \bar j_k'\}$ is a sequence of bundle ismorphisms $\theta_j : \xi_j \cong \xi_k'$ for $j \geq N$ , $N$ some integer, which are compatible with $\bar j_k$ and $\bar j_k'$ .

Here are some important examples:

A rank $j$ -vector $V \to X$ over a CW complex $X$ defines a stable vector bundle $\{ V \}$ by setting $X_k = X$ fixed for all $k$ , $\xi_k = 0$ for $k < j$ and $\xi_k = E \oplus \underline{\Rr}^{j-k}$ for $k\geq j$ with $\bar j_k$ the obvious inclusions.
If $M$ is a compact n-manifold then $M$ embedds into $\Rr^{2n+k}$ for $k \geq 0$ and for $k \geq 1$ two such embeddings are isotopic. The normal bundle of any such embedding defines a stable vector bundle over $M$ 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 $\nu_M$ for the stable normal bundle of $M$ . Note that by defintion $\nu_M$ is a stable inverse to the tangent bundle of $M$ :

$\displaystyle \nu_M \oplus TM \cong \{ \underline{\Rr} \}.$ $\displaystyle \nu_M \oplus TM \cong \{ \underline{\Rr} \}.$

The universal bundle $\gamma \to BO$ is the stable bundle with $X_k = BO(k)$ , $\xi_k$ the universal $k$ -plane bundle and $\bar j_k$ the classifymap of $\xi_k \oplus \underline{\Rr}$ .
If $E = \{ \xi_k, \bar j_k \}$ is a stable vector bundle over $\{ Y_k \}$ and $f$ is a sequence of maps $f_k : X_k \to Y_k$ compatible with inclusions then $\{ f_k^* \xi_k, f_k^* \bar j_k \}$ is a stable vector bundle over $\{X_k\}$ called the pull-back bundle and denoted $f^*E$ .

3 B-manifolds and B-bordism

Let $E = \{ \xi_k, X_k \}$ be a stable vector bundle. A normal $E$ -manifold is a triple $(M, f, \alpha)$ consisting of a compact manifold $M$ , a map sequence of maps $f_k : M \to X_k$ , and a stable bundle isomorphism $\alpha : f^* E \cong \nu_M$ .

Recall that $\xi: B \to BO$ is a fibration. We next define normal $B$ -manifolds and normal $B$ -bordism: more details are given in [Stong1968][Chapter II] and [Kreck1999][Section 1]. If $W$ is a compact smooth manifold then $\nu_W$ is classified by a map, also denoted $\nu_M$ , $\nu_W : W \to BO$ : $\nu_W^*\gamma \cong \nu_W$ . A $B$ -structure on $W$ is a map $\bar \nu: W \to B$ lifting $\nu_W$ through $B$ :

$\displaystyle \xymatrix{ & B \ar[d]^{\gamma} \\ W \ar[r]_{\nu_W}\ar[ur]^{\bar \nu} & BO.}$

A $B$ -manifold is a pair $(W, \bar \nu)$ where $\bar \nu$ is a $B$ -structure on $W$ . If $W$ has boundary $\partial W = M_0 \sqcup M_1$ , a disjoint union of two closed $(n-1)$ -manifolds with inclusions $i_\epsilon : M_\epsilon \to W$ , $\epsilon = 0, 1$ , then $W$ is a $B$ -bordism between the $B$ -manifolds $(M_0, \bar \nu \circ i_0)$ and $(M, \bar \nu \circ i_1)$ .

4 The Pontrjagin Thom isomorphism

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

5 Spectral sequences

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

6 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
[Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010

B-Bordism

Revision as of 19:04, 18 January 2010

Contents

1 Introduction

2 Stable vector bundles

3 B-manifolds and B-bordism

4 The Pontrjagin Thom isomorphism

5 Spectral sequences

6 References

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@@ Line 2: / Line 2: @@
 == 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 $U \to O$.
+In this article we recall the definition of the bordism groups of closed manifolds, ususally with extra topological structure: orientation, spin-structure, weak complex structure etc.
-Associated to each fibration $B$ there is a bordism group of closed n-manifolds with $B$ structure denoted
+We specify the extra structure by a fibration $\xi : B \to BO$ where
-$$\Omega_n^B ~~\text{or}~~ \Omega_n^G.$$
+$BO$ denotes the classifying space of the stable orthogonal group and $B$ is homotopy equivalent to a CW complex of finite type. For example the fibrations $BSO \to BO$
-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 these groups.
+and $BU \to BO$ induced by the canonical homomorphisms $SO \to O$ and $U \to O$ are used for respectively oriented and complex bordism.
+Abusing notation, one writes $B$ for the fibraion $\xi$: for each $B$ and $n \geq 0$ there is a bordism group of closed n-manifolds with $B$ structure denoted
+$$\Omega_n^B ~~\text{or}~~ \Omega_n(B).$$
+If $B = BG$ is the classifying space of the stable group $G$ then on often writes
+$$ \Omega_n^G ~~\text{or}~~ \Omega_n(G).$$
+</wikitex>
+== Stable vector bundles ==
+<wikitex>;
+We recall the defintion of a stable vector bundle from {{cite|Kreck&Lück2005|18.10}} which requires some notation.  Let $\underline{\Rr^k}$ denote the trivial rank k bundle $Z \times \Rr^k$: the space $Z$ will be clear from context.  Let $\{ X_k \}$ be a sequence $X_0 \to X_1 \to X_2 \to \dots$ of inclusions $j_k : X_k \to X_{k+1}$ of CW-complexes.  A stable vector bundle $E$ over $\{ X_k \}$ is a sequence of rank k-vector bundles $\xi_k \to X_k$ and a sequence of bundle maps
+$$ \bar j_k : \xi_k \oplus \underline{\Rr} \to \xi_{k+1}$$
+where $\bar j_k$ covers the inclusion $j_k$.  An isomorphism of stable vector bundles over $\{ X \}$, $\theta : \{ (\xi_k\, \bar j_k) \} \cong \{\xi_k', \bar j_k'\}$ is a sequence of bundle ismorphisms $\theta_j : \xi_j \cong \xi_k'$ for $j \geq N$, $N$ some integer, which are compatible with $\bar j_k$ and $\bar j_k'$.
+Here are some important examples:
+* A rank $j$-vector $V \to X$ over a CW complex $X$ defines a stable vector bundle $\{ V \}$ by setting $X_k = X$ fixed for all $k$, $\xi_k = 0$ for $k < j$ and $\xi_k = E \oplus \underline{\Rr}^{j-k}$ for $k\geq j$ with $\bar j_k$ the obvious inclusions.
+* If $M$ is a compact n-manifold then $M$ embedds into $\Rr^{2n+k}$ for $k \geq 0$ and for $k \geq 1$ two such embeddings are isotopic.  The normal bundle of any such embedding defines a stable vector bundle over $M$ 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 $\nu_M$ for the stable normal bundle of $M$.  Note that by defintion $\nu_M$ is a stable inverse to the tangent bundle of $M$:
+$$ \nu_M \oplus TM \cong \{ \underline{\Rr} \}.$$
+* The universal bundle $\gamma \to BO$ is the stable bundle with $X_k = BO(k)$, $\xi_k$ the universal $k$-plane bundle and $\bar j_k$ the classifymap of $\xi_k \oplus \underline{\Rr}$.
+* If $E = \{ \xi_k, \bar j_k \}$ is a stable vector bundle over $\{ Y_k \}$ and $f$ is a sequence of maps $f_k : X_k \to Y_k$ compatible with inclusions then $\{ f_k^* \xi_k, f_k^* \bar j_k \}$ is a stable vector bundle over $\{X_k\}$ called the pull-back bundle and denoted $f^*E$.
 </wikitex>
 == B-manifolds and B-bordism ==
-<wikitex>
+<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 $B$:
+Let $E = \{ \xi_k, X_k \}$ be a stable vector bundle.  A normal $E$-manifold is a triple $(M, f, \alpha)$ consisting of a compact manifold $M$, a map sequence of maps $f_k : M \to X_k$, and a stable bundle isomorphism $\alpha : f^* E \cong \nu_M$.
+Recall that $\xi: B \to BO$ is a fibration. We next define normal $B$-manifolds and normal $B$-bordism: more details are given in {{cite|Stong1968}}[Chapter II] and {{cite|Kreck1999}}[Section 1].  If $W$ is a compact smooth manifold then $\nu_W$ is classified by a map, also denoted $\nu_M$, $\nu_W : W \to BO$: $\nu_W^*\gamma \cong \nu_W$.  A $B$-structure on $W$ is a map $\bar \nu: W \to B$ lifting $\nu_W$ through $B$:
 $$
 \xymatrix{
   & B \ar[d]^{\gamma} \\
-M \ar[r]_{\nu_M}\ar[ur]^{\bar \nu} & BO}
+W \ar[r]_{\nu_W}\ar[ur]^{\bar \nu} & BO.}
 $$
+A $B$-manifold is a pair $(W, \bar \nu)$ where $\bar \nu$ is a $B$-structure on $W$.  If $W$ has boundary $\partial W = M_0 \sqcup M_1$, a disjoint union of two closed $(n-1)$-manifolds with inclusions $i_\epsilon : M_\epsilon \to W$, $\epsilon = 0, 1$, then $W$ is a $B$-bordism between the $B$-manifolds $(M_0, \bar \nu \circ i_0)$ and $(M, \bar \nu \circ i_1)$.
 </wikitex>