B-Bordism
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{{endthm}} | {{endthm}} | ||
− | 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: | + | 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: a specific isomorphism, up to appropriate equivalence, is required to give a bijection between the set of $B_r$ structures. Happily this is the case for the normal bundle of an embedding 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 | 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 | ||
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{{endthm}} | {{endthm}} | ||
− | 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 $(M \times [0, 1], \bar \nu_{M \times [0, 1]}$. | + | 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 $(M \times [0, 1], \bar \nu_{M \times [0, 1]})$. |
{{beginthm|Definition}} | {{beginthm|Definition}} | ||
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== The Pontrjagin Thom isomorphism == | == The Pontrjagin Thom isomorphism == | ||
<wikitex>; | <wikitex>; | ||
− | 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 {{cite|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 $T(E_r)$. | + | 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 {{cite|Kreck&Lück2005|18.10}} with $E_r \to B_r$ equal to the pullback bundle $\gamma_r^*(EO(r))$ where $EO(r)$ is the universal r-plane bundle over $BO(r)$. 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 | + | 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 we 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 increases these maps are compatibly related by suspension and the structure maps of the spectrum $MB$. Hence we obtain a homotopy class |
+ | $$P((M, \bar \mu)) \in lim_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$$ | ||
+ | The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$. | ||
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
There is an isomorphism of abelian groups | There is an isomorphism of abelian groups | ||
− | $$ | + | $$ P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto PT([M, \bar \nu]).$$ |
+ | {{endthm}} | ||
+ | For example, if $B_r = pt$ is the one-point space for each r, then $MB$ is the sphere spectrum $S$ and $\pi_n(S) = \pi_n^S$ is the n-th stable homotopy group. On the other hand, in this case $\Omega_n^B = \Omega_n^{fr}$ is the framed bordism group and as special case we have | ||
+ | {{beginthm|Theorem}} | ||
+ | There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$. | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> |
Revision as of 19:12, 21 January 2010
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Contents |
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
In this section we give a compressed accont of [Stong1968, Chapter II]. 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 .
Note that if and are isomorphic vector bundles over then the sets of -structures on each are in bijective equivalence. However -structures are defined on specific bundles, not isomorphism classes of bundles: a specific isomorphism, up to appropriate equivalence, is required to give a bijection between the set of structures. Happily this is the case for the normal bundle of an embedding as we now explain.
Let be a compact manifold and let be an embedding. Equipping with the standard metric, the normal bundle of is a rank r vector bundle over classified by its normal Gauss map . If is another such embedding and , then is regularly homotopic to and all regular homotopies are regularly homotopic relative to their endpoints. A regular homotopy defines an isomorphism and a regular homotopy of regular homotopies gives a homotopy between these isomorphisms. Taking care one proves the following
Lemma 2.2 [Stong1968, p 15]. For r sufficiently large, (depending only on n) there is a 1-1 correspondence between the structures of the normal bundles of any two embeddings .
Now let be a sequence of fibrations over with maps fitting into the following commutative diagram
where is the standard inclusion and let . A -structure on the normal bundle of an embedding defines a unique -structure on the composition of with the standard inclusion .
Defition 2.3. A -structure on is an equivalence class of -structure on where two such structures are equivalent if they become equivalent for r sufficiently large. A -manifold is a pair where is a compact manifold and is a -structure on .
If is a compact manifold with boundary then by choosing the inward-pointing normal vector along , a -structure on restricts to a -structure on . In particular, if is a closed manifold then has a canonical -structure such that restricting to on . The restriction of this -structure to is denoted : by construction is the boundary of .
Definition 2.4. Closed -manifolds and are -bordant if there is a compact -manifold such that . We write for the bordism class of .
Proposition 2.5. The set of -borism class of closed n-manifolds with -structure,
forms an abelian group under the operation of disjoint union with inverse .
3 The Pontrjagin Thom isomorphism
If is a vector bundle, let denote its Thom space. Recall that is a sequence of fibrations with compatible maps . These data give rise to a stable vector bundle as defined in [Kreck&Lück2005, 18.10] with equal to the pullback bundle where is the universal r-plane bundle over . This stable vector bundle defines a Thom spectrum which we denote . The r-th space of is .
By definition a -manifold is an equivalence class of -structures on , the normal bundle of an embedding . Hence gives rise to the collapse map where we identify with the one-point compatificiation of , we map via on a tubular neighbourhood of and we map all other points to the base-point of . As r increases these maps are compatibly related by suspension and the structure maps of the spectrum . Hence we obtain a homotopy class
The celebrated theorem of Pontrjagin and Thom states in part that depends only on the bordism class of .
Theorem 3.1. There is an isomorphism of abelian groups
For example, if is the one-point space for each r, then is the sphere spectrum and is the n-th stable homotopy group. On the other hand, in this case is the framed bordism group and as special case we have
Theorem 3.2. There is an isomorphism .
4 Spectral sequences
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