B-Bordism

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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 $\gamma : B \to BO$${{Stub}} == Introduction == <wikitex>; On this page we recall the definition of the bordism groups of closed manifolds, with extra topological structure: orientation, spin-structure, weak complex structure etc. The ideas here date back to {{cite|Lashof1965}}. There is a deatiled treatment in {{cite|Stong1968|Chapter II}} and a summary in {{cite|Kreck1999|Section 1}} as well as {{cite|Kreck&Lück2005|18.10}}. We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$ where $BO$ denotes the classifying space of the stable orthogonal group and $B$ is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes $B$ for the fibraion $\gamma$. A $B$-manifold is a compact manifold $M$ together with a lift of a classifying map for the stable normal bundle of $M$ to $B$: $$ \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>; 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)$. {{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 $\gamma_r \circ \bar \xi = \xi$. {{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: 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 {{beginthm|Lemma|{{cite|Stong1968|p 15}}}} For r sufficiently large, (depending only on n) there is a 1-1 correspondence between the $B_r$ structures of the normal bundles of any two embeddings $i_0, i_1 : M \to \Rr^{n+r}$. {{endthm}} 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 $$ \xymatrix{ B_r \ar[r]^{g_r} \ar[d]^{\gamma_r} & B_{r+1} \ar[d]^{\gamma_{r+1}} \ BO(r) \ar[r]^{j_r} & BO(r+1) } $$ where $j_r$ is the standard inclusion and let $B = lim_{r \to \infty}(B_r)$. A $B_r$-structure on the normal bundle of an embedding $i: M \to \Rr^{n+r}$ defines a unique $B_{r+1}$-structure on the composition of $i$ with the standard inclusion $\Rr^{n+r} \to \Rr^{n+r+1}$. {{beginthm|Defition|{{cite|Stong1968|p 15}}}} A $B$-structure on $M$ is an equivalence class of $B_r$-structure on $M$ where two such structures are equivalent if they become equivalent for r sufficiently large. A $B$-manifold is a pair $(M, \bar \nu)$ where $M$ is a compact manifold and $\bar \nu$ is a $B$-structure on $M$. {{endthm}} If $W$ is a compact manifold with boundary $\partial W$ then by choosing the inward-pointing normal vector along $\partial W$, a $B$-structure on $W$ restricts to a $B$-structure on $\partial W$. In particular, if $(M, \bar \nu_M)$ is a closed $B$ manifold then $W = M \times [0, 1]$ has a canonical $B$-structure $\bar \nu_{M \times [0, 1]}$ such that restricting to $(M, \bar \nu_M)$ on $M \times \{ 0 \}$. The restriction of this $B$-structure to $M \times \{ 1 \}$ is denoted $-\bar \nu$: by construction $(M \sqcup M, \bar \nu \sqcup - \bar \nu)$ is the boundary of $(M \times [0, 1], \bar \nu_{M \times [0, 1]})$. {{beginthm|Definition}} Closed $B$-manifolds $(M_0, \bar \nu_0)$ and $(M_1, \bar \nu_1)$ are $B$-bordant if there is a compact $B$-manifold $(W, \bar \nu)$ such that $\partial(W, \bar \nu) = (M_0 \sqcup M_1, \bar \nu_0 \sqcup -\bar \nu_1)$. We write $[M, \bar \nu]$ for the bordism class of $(M, \bar \nu)$. {{endthm}} {{beginthm|Proposition|{{cite|Stong1968|p 17}}}} The set of $B$-borism class of closed n-manifolds with $B$-structure, $$ \Omega_n^B := \{ [M, \bar \nu ] \},$$ forms an abelian group under the operation of disjoint union with inverse $-[M,\bar \nu] = [M, -\bar \nu]$. {{endthm}} </wikitex> == The Pontrjagin Thom isomorphism == <wikitex>; If $E$ is a vector bundle, let $T(E)$ denote its Thom space. Recall that $B = (B_r, \gamma_r, g_r)$ is a sequence of fibrations $\gamma_r : B_r \to BO(r)$ with compatible maps $g_r : B_r \to B_{r+1}$. These data give rise to a stable vector bundle as defined in {{cite|Kreck&Lück2005|18.10}} with $E_r \to B_r$ equal to the pullback bundle $\gamma_r^*(EO(r))$ where $EO(r)$ is the universal r-plane bundle over $BO(r)$. This stable vector bundle defines a Thom spectrum which we denote $MB$. The r-th space of $MB$ is $T(E_r)$. By definition a $B$-manifold $(M, \bar \nu)$ is an equivalence class of $B_r$-structures on $\nu(i)$, the normal bundle of an embedding $i : M \to \Rr^{n+r}$. Hence $(M, \bar \nu)$ gives rise to the collapse map $S^{n+r} \to T(E_r)$ where we identify $S^{n+r}$ with the one-point compatificiation of $\Rr^{n+r}$, we map via $\bar \nu_r$ on a tubular neighbourhood of $i(M) \subset \Rr^{n+r}$ and we map all other points to the base-point of $T(E_r)$. As r increases these maps are compatibly related by suspension and the structure maps of the spectrum $MB$. Hence we obtain a homotopy class $$P((M, \bar \mu)) \in lim_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$$ The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$. {{beginthm|Theorem}} There is an isomorphism of abelian groups $$ P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto PT([M, \bar \nu]).$$ {{endthm}} For example, if $B_r = pt$ is the one-point space for each r, then $MB$ is the sphere spectrum $S$ and $\pi_n(S) = \pi_n^S$ is the n-th stable homotopy group. On the other hand, in this case $\Omega_n^B = \Omega_n^{fr}$ is the framed bordism group and as special case we have {{beginthm|Theorem}} There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$. {{endthm}} </wikitex> == Spectral sequences == <wikitex>; $$ H_p(B; \pi_q^S) \Longrightarrow \Omega_{p+q}^B$$ </wikitex> == References == {{#RefList:}} [[Category:Theory]]\gamma : B \to BO$ where $BO$$BO$ denotes the classifying space of the stable orthogonal group and $B$$B$ is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes $B$$B$ for the fibraion $\gamma$$\gamma$. A $B$$B$-manifold is a compact manifold

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$M$ together with a lift of a classifying map for the stable normal bundle of

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$M$ to $B$$B$:

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

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

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

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

2 B-structures

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

Definition 2.1. Let $\xi: E \to X$$\xi: E \to X$ be a rank r vector bundle classified by $\xi : X \to BO(r)$$\xi : X \to BO(r)$. A $B_r$$B_r$-structure on $\xi$$\xi$ is a vertical homotopy class of maps $\bar \xi : X \to B_r$$\bar \xi : X \to B_r$ such that $\gamma_r \circ \bar \xi = \xi$$\gamma_r \circ \bar \xi = \xi$.

Note that if $\xi_0$$\xi_0$ and $\xi_1$$\xi_1$ are isomorphic vector bundles over $X$$X$ then the sets of $B_r$$B_r$-structures on each are in bijective equivalence. However $B_r$$B_r$-structures are defined on specific bundles, not isomorphism classes of bundles: a specific isomorphism, up to appropriate equivalence, is required to give a bijection between the set of $B_r$$B_r$ structures. Happily this is the case for the normal bundle of an embedding as we now explain.

Let

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

Lemma 2.2 [Stong1968, p 15]. For r sufficiently large, (depending only on n) there is a 1-1 correspondence between the $B_r$$B_r$ structures of the normal bundles of any two embeddings $i_0, i_1 : M \to \Rr^{n+r}$$i_0, i_1 : M \to \Rr^{n+r}$.

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

$\displaystyle \xymatrix{ B_r \ar[r]^{g_r} \ar[d]^{\gamma_r} & B_{r+1} \ar[d]^{\gamma_{r+1}} \\ BO(r) \ar[r]^{j_r} & BO(r+1) }$

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

Defition 2.3 [Stong1968, p 15].

A $B$$B$-structure on

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$M$ is an equivalence class of $B_r$$B_r$-structure on

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$M$ where two such structures are equivalent if they become equivalent for r sufficiently large. A $B$$B$-manifold is a pair $(M, \bar \nu)$$(M, \bar \nu)$ where

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$M$ is a compact manifold and $\bar \nu$$\bar \nu$ is a $B$$B$-structure on

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$M$.

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

Definition 2.4. Closed $B$$B$-manifolds $(M_0, \bar \nu_0)$$(M_0, \bar \nu_0)$ and $(M_1, \bar \nu_1)$$(M_1, \bar \nu_1)$ are $B$$B$-bordant if there is a compact $B$$B$-manifold $(W, \bar \nu)$$(W, \bar \nu)$ such that $\partial(W, \bar \nu) = (M_0 \sqcup M_1, \bar \nu_0 \sqcup -\bar \nu_1)$$\partial(W, \bar \nu) = (M_0 \sqcup M_1, \bar \nu_0 \sqcup -\bar \nu_1)$. We write $[M, \bar \nu]$$[M, \bar \nu]$ for the bordism class of $(M, \bar \nu)$$(M, \bar \nu)$.

Proposition 2.5 [Stong1968, p 17]. The set of $B$$B$-borism class of closed n-manifolds with $B$$B$-structure,

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

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

3 The Pontrjagin Thom isomorphism

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

By definition a $B$$B$-manifold $(M, \bar \nu)$$(M, \bar \nu)$ is an equivalence class of $B_r$$B_r$-structures on $\nu(i)$$\nu(i)$, the normal bundle of an embedding $i : M \to \Rr^{n+r}$$i : M \to \Rr^{n+r}$. Hence $(M, \bar \nu)$$(M, \bar \nu)$ gives rise to the collapse map $S^{n+r} \to T(E_r)$$S^{n+r} \to T(E_r)$ where we identify $S^{n+r}$$S^{n+r}$ with the one-point compatificiation of $\Rr^{n+r}$$\Rr^{n+r}$, we map via $\bar \nu_r$$\bar \nu_r$ on a tubular neighbourhood of $i(M) \subset \Rr^{n+r}$$i(M) \subset \Rr^{n+r}$ and we map all other points to the base-point of $T(E_r)$$T(E_r)$. As r increases these maps are compatibly related by suspension and the structure maps of the spectrum $MB$$MB$. Hence we obtain a homotopy class

$\displaystyle P((M, \bar \mu)) \in lim_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$ $$\displaystyle P((M, \bar \mu)) \in lim_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$$

The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$$P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$$(M, \bar \nu)$.

Theorem 3.1. There is an isomorphism of abelian groups

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

For example, if $B_r = pt$$B_r = pt$ is the one-point space for each r, then $MB$$MB$ is the sphere spectrum $S$$S$ and $\pi_n(S) = \pi_n^S$$\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}$$\Omega_n^B = \Omega_n^{fr}$ is the framed bordism group and as special case we have

Theorem 3.2. There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$$P : \Omega_n^{fr} \cong \pi_n^S$.

4 Spectral sequences

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

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