B-Bordism

Revision as of 18:58, 21 January 2010

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 ${{Stub}} == Introduction == <wikitex>; 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 {{cite|Lashof1965}}. There is a deatiled treatment in {{cite|Stong1968|Chapter II}} and a summary in {{cite|Kreck1999|Section 1}} as well as {{cite|Kreck&Lück2005|18.10}}. We specify extra topological structure universally by means of a fibration $\xi : B \to BO$ where $BO$ denotes the classifying space of the stable orthogonal group and $B$ is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes $B$ for the fibraion $\xi$. A $B$-manifold is a compact manifold $M$ together with a lift of a classifying map for the stable normal bundle of $M$ to $B$: $$ \xymatrix{ & B \ar[d]^{\gamma} \ W \ar[r]_{\nu_W}\ar[ur]^{\bar \nu} & BO.} $$ The n-dimensional $B$-bordism group is defined to be the set of closed $B$-manifolds up modulo the relation of $B$-bordism and addition given by disjoint union $$ \Omega_n^B := \{ [M, \bar \nu] \}/\simeq .$$ Alternative notations are $\Omega_n(B)$ and also $\Omega_n^G$ when $B \to BO = BG \to BO$ for $G \to O$ a stable represenation of a topological group $G$. Details of the definition and some important theorems for computing $\Omega_n^B$ follow. </wikitex> == B-structures == <wikitex>; Let $G_{r, m}$ denote the Grassman manifold of unoriented r-planes in $\Rr^n$ and let $BO(r) = lim_{m \to \infty} G_{r, m}$ be the infinite Grassman and fix a fibration $\xi_r : B_r \to BO(r)$. {{beginthm|Definition}} Let $\xi: E \to X$ be a rank r vector bundle classified by $\xi : X \to BO(r)$. A $B_r$-structure on $\xi$ is a vertical homotopy class of maps $\bar \xi : X \to B_r$ such that $\xi_r \circ \bar \xi = \xi$. {{endthm}} </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>; 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 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 $B$: $$ \xymatrix{ & B \ar[d]^{\gamma} \ W \ar[r]_{\nu_W}\ar[ur]^{\bar \nu} & BO.} $$ 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> == 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]]\gamma : 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. Abusing notation, one writes $B$ for the fibraion $\gamma$. A $B$-manifold is a compact manifold $M$ together with a lift of a classifying map for the stable normal bundle of $M$ to $B$:

The n-dimensional $B$-bordism group is defined to be the set of closed $B$-manifolds up modulo the relation of $B$-bordism and addition given by disjoint union

$$\displaystyle \Omega_n^B := \{ (M, \bar \nu) \}/\simeq.$$

Alternative notations are $\Omega_n(B)$ and also $\Omega_n^G$ when $B \to BO = BG \to BO$ for $G \to O$ a stable represenation of a topological group $G$. Details of the definition and some important theorems for computing $\Omega_n^B$ follow.

2 B-structures

In this section we give a compressed accont of [Stong1968, Chapter II]. Let $G_{r, m}$ denote the Grassman manifold of unoriented r-planes in $\Rr^n$ and let $BO(r) = lim_{m \to \infty} G_{r, m}$ be the infinite Grassman and fix a fibration $\gamma_r : B_r \to BO(r)$.

Note that if $\xi_0$ and $\xi_1$ are isomorphic vector bundles over $X$ then the sets of $B_r$-structures on each are in bijective equivalence. However $B_r$-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 $B_r$ structures. Happily this is the case for the normal bundle as we now explain.

Let $M$ be a compact manifold and let $i_0 : M \to \Rr^{n+r}$ be an embedding. Equipping $\Rr^{n+r}$ with the standard metric, the normal bundle of $i_0$ is a rank r vector bundle over classified by its normal Gauss map $\nu(i_0) : M \to G_{r, n+r} \subset BO(r)$. If $i_1$ is another such embedding and $r >> n$, then $i_1$ is regularly homotopic to $i_0$ and all regular homotopies are regularly homotopic relative to their endpoints. A regular homotopy $H$ defines an isomorphism $\alpha_H :\nu(i_0) \cong \nu(i_1)$ and a regular homotopy of regular homotopies gives a homotopy between these isomorphisms. Taking care one proves the following

Now let $(B_r, \gamma_r)$ be a sequence of fibrations over $BO(r)$ with maps $g_r : B_r \to B_{r+1}$ fitting into the following commutative diagram

where $j_r$ is the standard inclusion and let $B = lim_{r \to \infty}(B_r)$. A $B_r$-structure on the normal bundle of an embedding $i: M \to \Rr^{n+r}$ defines a unique $B_{r+1}$-structure on the composition of $i$ with the standard inclusion $\Rr^{n+r} \to \Rr^{n+r+1}$.

If $W$ is a compact manifold with boundary $\partial W$ then by choosing the inward-pointing normal vector along $\partial W$, a $B$-structure on $W$ restricts to a $B$-structure on $\partial W$. In particular, if $(M, \bar \nu_M)$ is a closed $B$ manifold then $W = M \times [0, 1]$ has a canonical $B$-structure $\bar \nu_{M \times [0, 1]}$ such that restricting to $(M, \bar \nu_M)$ on $M \times \{ 0 \}$. The restriction of this $B$-structure to $M \times \{ 1 \}$ is denoted $-\bar \nu$: by construction $(M \sqcup M, \bar \nu \sqcup - \bar \nu)$ is the boundary of $(M \times [0, 1], \bar \nu_{M \times [0, 1]}$.

Proposition 2.5. The set of $B$-borism class of closed n-manifolds with $B$-structure,

$$\displaystyle \Omega_n^B := \{ [M, \bar \nu ] \},$$

forms an abelian group under the operation of disjoint union with inverse $-[M,\bar \nu] = [M, -\bar \nu]$.

3 The Pontrjagin Thom isomorphism

If $E$ is a vector bundle, let $T(E)$ denote its Thom space. Recall that $B = (B_r, \gamma_r, g_r)$ is a sequence of fibrations $\gamma_r : B_r \to BO(r)$ with compatible maps $g_r : B_r \to B_{r+1}$. These data give rise to a stable vector bundle as defined in [Kreck&Lück2005, 18.10] with $E_r \to B_r$ equal to the pullback bundle $\gamma_r^*(EO(r))$ where $EO(r)$. This stable vector bundle defines a Thom spectrum which we denote $MB$. The r-th space of $MB$ is $T(E_r)$.

By definition a $B$-manifold $(M, \bar \nu)$ is an equivalence class of $B_r$ structures on $\nu(i)$, the normal bundle of an embedding $i : M \to \Rr^{n+r}$. Hence $(M, \bar \nu)$ gives rise to the collapse map $S^{n+r} \to T(E_r)$ where identify $S^{n+r}$ with the one-point compatificiation of $\Rr^{n+r}$, we map via $\bar \nu_r$ on a tubular neighbourhood of $i(M) \subset \Rr^{n+r}$ and we map all other points to the base-point of $T(E_r)$. As r increase these maps are compatibly related by suspension and the structure maps of the spectrum $MB$. Hence we obtain a homotopy class $PT((M, \bar \mu)) \in lim_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB)$. The celebrated theorem of Pontrjagin and Thom states in part that $PT(M, \bar \nu)$ depends only on the bordism class $[M, \bar]$.

Theorem 3.1. There is an isomorphism of abelian groups

$$\displaystyle PT : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto PT([M, \bar \nu]).$$

4 Spectral sequences

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

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