B-Bordism
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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 are detailed treatment in {{cite|Stong1968|Chapter II}} and summaries in {{cite|Kreck&Lück2005|18.10}} and {{cite|Kreck1999|Section 1}}. | 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 are detailed treatment in {{cite|Stong1968|Chapter II}} and summaries in {{cite|Kreck&Lück2005|18.10}} and {{cite|Kreck1999|Section 1}}. | ||
− | We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$ 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 fibration $\gamma$. | + | We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$ 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 fibration $\gamma$. Speaking somewhat imprecisely (precise details are below) 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{ | \xymatrix{ | ||
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W \ar[r]_{\nu_W} \ar[ur]^{\bar \nu} & BO.} | 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 $B$- | + | The n-dimensional $B$-bordism group is defined to be the set of closed $B$-manifolds up modulo the relation of bordism via compact $B$-manifolds. Addition given by disjoint union and in fact for each $n >0$ there is a group |
$$ \Omega_n^B := \{ (M, \bar \nu) \}/\simeq.$$ | $$ \Omega_n^B := \{ (M, \bar \nu) \}/\simeq.$$ | ||
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− | == B-structures == | + | == B-structures and bordisms == |
<wikitex>; | <wikitex>; | ||
− | In this section we give a compressed accont of {{cite|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 $\gamma_r : B_r \to BO(r)$. | + | In this section we give a compressed accont of parts of {{cite|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 $\gamma_r : B_r \to BO(r)$. |
{{beginthm|Definition}} | {{beginthm|Definition}} | ||
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{{endthm}} | {{endthm}} | ||
− | + | This lemma is one motivataion for the useful, but subtle, notion of a fibred stable vector bundle. | |
+ | |||
+ | {{beginthm|Definition}} | ||
+ | A fibred stable vector bundle $B$ consists of the following data: sequence of fibrations over $BO(r)$, $(B_r, \gamma_r)$ and a sequence of maps $g_r : B_r \to B_{r+1}$ fitting into the following commutative diagram | ||
$$ | $$ | ||
\xymatrix{ | \xymatrix{ | ||
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} | } | ||
$$ | $$ | ||
− | where $j_r$ is the standard inclusion and let $B = lim_{r \to \infty}(B_r)$. | + | where $j_r$ is the standard inclusion and let $B = lim_{r \to \infty}(B_r)$. |
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Remark}} | ||
+ | A fibred stable vector bundle $B$ gives rise to a stable vector bundle as defined in {{cite|Kreck&Lück2005|18.10}}. One defines $E_r \to B_r$ to be the pullback bundle $\gamma_r^*(EO(r))$ where $EO(r)$ is the universal r-plane bundle over $BO(r)$. Then the maps $g_r$ define bundle maps $\bar g_r : E_r \times \Rr \to E_{r+1}$ covering the maps $g_r$. | ||
+ | {{endthm}} | ||
+ | |||
+ | Now 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 $\Rr^{n+r} \to \Rr^{n+r+1}$. Hence we can make the following | ||
− | {{beginthm| | + | {{beginthm|Definition|{{cite|Stong1968|p 15}}}} |
− | A $B$-structure on $M$ is an equivalence class of $B_r$-structure on $M$ where two such structures are equivalent if they become equivalent for r sufficiently large. A $B$-manifold is a pair $(M, \bar \nu)$ where $M$ is a compact manifold and $\bar \nu$ is a $B$-structure on $M$. | + | Let $B$ be a fibred stable vectore bundle. A $B$-structure on $M$ is an equivalence class of $B_r$-structure on $M$ where two such structures are equivalent if they become equivalent for r sufficiently large. A $B$-manifold is a pair $(M, \bar \nu)$ where $M$ is a compact manifold and $\bar \nu$ is a $B$-structure on $M$. |
{{endthm}} | {{endthm}} | ||
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− | == | + | === Singular bordism === |
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− | If $ | + | $B$-bordism gives rise a generalised homology theory. If $X$ is a space then the n-cycles of this homology theory are pairs |
+ | $$ ((M, \bar \nu),~ f: M \to X) $$ | ||
+ | where $(M, \bar \nu)$ is a closed n-dimensional $B$-manifold and $f$ is any continuous map. Two cycles $((M_0, \bar \nu_0), f_0)$ and $((M_1, \bar \nu_1), f_1)$ are homologous if there is a pair | ||
+ | $$ ((W, \bar \nu),~ g : W \to X)$$ | ||
+ | where $(W, \bar \nu)$ is a $B$-bordism from $(M_0, \bar \nu_0)$ to $(M_1, \bar \nu_1)$ and $g : W \to X$ is a continuous map extending $f_0 \sqcup f_1$. Writing $[(M, \bar \nu), f]$ for the equivalence class of $((M, \bar \nu) ,f)$ we obtain an abelian group | ||
+ | $$ \Omega_n^B(X) : = \{ [(M, \bar \nu), f] \} $$ | ||
+ | with group operation disjoint union and inverse $-[(M, \bar \nu), f] = [(M, - \bar \nu), f]$. | ||
+ | {{beginthm|Proposition}} | ||
+ | The mapping $X \to \Omega_n^B(X)$ defines a generalised homology theory with coefficients $\Omega_n^B(pt) = \Omega_n^B$. | ||
+ | {{endthm}} | ||
+ | Given a stable vector bundle $B = B_r, \gamma_r, g_r)$ we can form the stable vectore bundle $B \times X := (B_r \times X, \gamma_r \times X, g_r \times \id_X)$. The following simple lemma is clear but often useful. | ||
+ | {{beginthm|Lemma}} | ||
+ | For any space $X$ there is an isomorphism $\Omega_n^B(X) \cong \Omega_n^{B \times X}$. | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
+ | |||
+ | == The Pontrjagin-Thom isomorphism == | ||
+ | <wikitex>; | ||
+ | If $E$ is a vector bundle, let $T(E)$ denote its Thom space. Recall that that a fibred stable vector bundle $B = (B_r, \gamma_r, g_r)$ defines a stable vector bundle $(E_r, \gamma_r, \bar g_r)$ where $E_r = \gamma_r^*(EO(r))$. 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 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 | + | 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 |
− | $$P((M, \bar \ | + | $$ c(M, \bar \nu) : S^{n+r} \to T(E_r)$$ |
+ | where we identify $S^{n+r}$ with the one-point compatificiation of $\Rr^{n+r}$, we map via $\bar \nu_r$ on a tubular neighbourhood of $i(M) \subset \Rr^{n+r}$ and we map all other points to the base-point of $T(E_r)$. As r increases these maps are compatibly related by suspension and the structure maps of the spectrum $MB$. Hence we obtain a homotopy class | ||
+ | $$ [c(M, \bar \nu)] =: P((M, \bar \nu)) \in lim_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$$ | ||
The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$. | The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$. | ||
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
There is an isomorphism of abelian groups | There is an isomorphism of abelian groups | ||
− | $$ P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto | + | $$ P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto P([M, \bar \nu]).$$ |
{{endthm}} | {{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 | 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}} | {{beginthm|Theorem}} | ||
There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$. | There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$. | ||
+ | {{endthm}} | ||
+ | The Pontrjagin-Thom isomorphism generalises to singular bordism. | ||
+ | {{beginthm|Theorem}} | ||
+ | For any space $X$ there is an isomorphism of abelian groups | ||
+ | $$ P : \Omega_n^B(X) \cong \pi_n^S(MB \wedge X)$$ | ||
+ | where $MB \wedge X$ denotes the smash produce of the specturm $MB$ and the space $X$. | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
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== Spectral sequences == | == Spectral sequences == | ||
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− | $$ H_p(B; \pi_q^S) \Longrightarrow \Omega_{p+q}^B$$ | + | For any generalised homology thoery there $h_*$ there is a spectral sequence, called the Atiyah-Hirzeburbh spectral sequence (AHSS) which can be used to compute. $h_*(X)$. The $E_2$ term of the AHSS is $H_p(X; h_q(pt)$ and one writes |
+ | $$ \bigoplus_{p+q = n} H_p(B; h_q(pt)) \Longrightarrow h_{n}(X).$$ | ||
+ | The Pontryagin-Thom isomorphisms above therefore give rise to the following theorems. For the first we recall that stable homotopy defines a generalised homology theory. | ||
+ | {{beginthm|Theorem}} | ||
+ | Let $B$ be a fibred stable vector bundle. There is a spectral sequence | ||
+ | $$ \bigoplus_{p+q = n} H_p(B; \pi_q^S) \Longrightarrow \Omega_{n}^B$$ | ||
+ | {{beginthm|Theorem}} | ||
+ | Let $B$ be a fibred stable vector bundle and $X$ a space. There is a spectral sequence | ||
+ | $$ \bigoplus_{p+q = n} H_p(B; \Omega_q^B) \Longrightarrow \Omega_n^B(X).$$ | ||
</wikitex> | </wikitex> | ||
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Theory]] | [[Category:Theory]] |
Revision as of 12:51, 25 January 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
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 are detailed treatment in [Stong1968, Chapter II] and summaries in [Kreck&Lück2005, 18.10] and [Kreck1999, Section 1].
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 fibration . Speaking somewhat imprecisely (precise details are below) 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 via compact -manifolds. Addition given by disjoint union and in fact for each there is a group
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 and bordisms
In this section we give a compressed accont of parts 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 .
This lemma is one motivataion for the useful, but subtle, notion of a fibred stable vector bundle.
Definition 2.3. A fibred stable vector bundle consists of the following data: sequence of fibrations over , and a sequence of maps fitting into the following commutative diagram
where is the standard inclusion and let .
Remark 2.4. A fibred stable vector bundle gives rise to a stable vector bundle as defined in [Kreck&Lück2005, 18.10]. One defines to be the pullback bundle where is the universal r-plane bundle over . Then the maps define bundle maps covering the maps .
Now a -structure on the normal bundle of an embedding defines a unique -structure on the composition of with the standard inclusion . Hence we can make the following
Definition 2.5 [Stong1968, p 15]. Let be a fibred stable vectore bundle. 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 2.6. Closed -manifolds and are -bordant if there is a compact -manifold such that . We write for the bordism class of .
Proposition 2.7 [Stong1968, p 17]. The set of -borism class of closed n-manifolds with -structure,
forms an abelian group under the operation of disjoint union with inverse .
2.1 Singular bordism
-bordism gives rise a generalised homology theory. If is a space then the n-cycles of this homology theory are pairs
where is a closed n-dimensional -manifold and is any continuous map. Two cycles and are homologous if there is a pair
where is a -bordism from to and is a continuous map extending . Writing for the equivalence class of we obtain an abelian group
with group operation disjoint union and inverse .
Proposition 2.8. The mapping defines a generalised homology theory with coefficients .
Given a stable vector bundle we can form the stable vectore bundle . The following simple lemma is clear but often useful.
Lemma 2.9. For any space there is an isomorphism .
3 The Pontrjagin-Thom isomorphism
If is a vector bundle, let denote its Thom space. Recall that that a fibred stable vector bundle defines a stable vector bundle where . 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 .
The Pontrjagin-Thom isomorphism generalises to singular bordism.
Theorem 3.3. For any space there is an isomorphism of abelian groups
where denotes the smash produce of the specturm and the space .
4 Spectral sequences
For any generalised homology thoery there there is a spectral sequence, called the Atiyah-Hirzeburbh spectral sequence (AHSS) which can be used to compute. . The term of the AHSS is and one writes
The Pontryagin-Thom isomorphisms above therefore give rise to the following theorems. For the first we recall that stable homotopy defines a generalised homology theory.
Theorem 4.1. Let be a fibred stable vector bundle. There is a spectral sequence
Theorem 4.2. Let be a fibred stable vector bundle and a space. There is a spectral sequence
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