B-Bordism
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1 Introduction
On this page we recall the definition of the bordism groups of closed smooth manifolds, with extra topological structure: orientation, spin-structure, weakly almost complex structure etc. The situation for piecewise linear and topological manifolds is similar and we discuss it briefly below.
The formulation of the general set-up for B-Bordism dates back to [Lashof1963]. There are detailed treatments in [Stong1968, Chapter II] and [Bröcker&tom Dieck1970] as well as summaries in [Kreck&Lück2005, 18.10] and [Kreck1999, Section 1]. See also the Wikipedia bordism page.
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 to
of a classifying map for the stable normal bundle of
:
![\displaystyle \xymatrix{ & B \ar[d]^{\gamma} \\ W \ar[r]_{\nu_W} \ar[ur]^{\bar \nu} & BO.}](/images/math/4/8/0/4801d251719f7177254c076b305ed742.png)
The -dimensional
-bordism group is defined to be the set of closed
-manifolds modulo the relation of bordism via compact
-manifolds. Addition is given by disjoint union and in fact for each
there is a group
![\displaystyle \Omega_n^B := \{ (M, \bar \nu) \}/\equiv.](/images/math/c/1/f/c1fd0f8829ed9828ee4208d8cc146ae5.png)
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.
1.1 Examples
We list some fundamental examples with common notation and also indicate the fibration B.
- Unoriented bordism:
;
.
- Oriented bordism:
,
;
.
- Spin bordism:
;
.
- Spin
bordism:
;
.
- String bodism :
;
.
- Complex bordism :
;
.
- Special unitary bordism :
;
.
- Framed bordism :
;
, the path space fibration.
2 B-structures and bordisms
In this section we give a compressed accont of parts of [Stong1968, Chapter II]. Let denote the Grassmann manifold of unoriented
-planes in
and let
be the infinite Grassmannian 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 sets 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 (see [Hirsch1959]). 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 set of structures of the normal bundles of any two embeddings
.
This lemma is one motivation 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: a sequence of fibrations
together with a sequence of maps
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) }](/images/math/7/c/1/7c1c615ec7ae50c39e222dd0f32582ad.png)
where is the standard inclusion. We 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
. The diagram above gives rise to bundle maps
covering the maps
: here
denotes the trivial rank 1 bundle over
.
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
which restricts 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 -bordism classes of closed n-manifolds with
-structure,
![\displaystyle \Omega_n^B := \{ [M, \bar \nu ] \},](/images/math/0/e/c/0ec9d94e5d00beba3be787adfd09d089.png)
forms an abelian group under the operation of disjoint union with inverse .
3 Singular bordism
-bordism gives rise to a generalised homology theory. If
is a space then the
-cycles of this homology theory are pairs
![\displaystyle ((M, \bar \nu),~ f: M \to X)](/images/math/4/9/6/49696267f7e79e5635fa29f69f16c1aa.png)
where is a closed
-dimensional
-manifold and
is any continuous map. Two cycles
and
are homologous if there is a pair
![\displaystyle ((W, \bar \nu),~ g : W \to X)](/images/math/3/2/e/32ee9740ebeffdfc531aa4ba26e04c96.png)
where is a
-bordism from
to
and
is a continuous map extending
. Writing
for the equivalence class of
we obtain an abelian group
![\displaystyle \Omega_n^B(X) : = \{ [(M, \bar \nu), f] \}](/images/math/0/e/9/0e94afc4f128e412222f4c91e415ee41.png)
with group operation disjoint union and inverse .
Proposition 3.1.
The mapping defines a generalised homology theory with coefficients
.
Given a stable vector bundle we can form the stable vector bundle
. The following simple lemma is clear but often useful.
Lemma 3.2.
For any space there is an isomorphism
.
4 The orientation homomorphism
We fix a local orientation at the base-point of . It then follows that every closed
-manifold
is given a local orientation. This amounts to a choice of fundamental class of
which is a generator
![\displaystyle [M] \in H_n(M; \underline{\Zz})](/images/math/c/3/e/c3e19cf09ded2cfda7d3aef03e7a2198.png)
where denotes the local coefficient system defined by the orientation character of
.
Given a closed -manifold
we can use
to push the fundamental class of
to
: now the local coefficient system is defined by the orientation character of the stable bundle
. It is easy to check that
denepnds only on the
-bordism class of
and is additive with respect to the operations
on
.
Definition 4.1.
Let be a fibred stable vector bundle. The orientation homomorphism is defined as follows:
![\displaystyle \rho : \Omega_n^B \to H_n(B; \underline{\Zz}), ~~~[M, \bar \nu] \mapsto \bar \nu_*[M].](/images/math/a/b/3/ab3eb885589c82b5a77ff4fccdf229bd.png)
For the singular bordism groups we have no bundle over
so in general there is only a
-valued orientation homomorphism. However, if the first Stiefel-Whitney class of
vanishes,
, then all
-manifolds are oriented in the usual sense and the orientation homomorphism can be lifted to
.
Definition 4.2.
Let be a fibred stable vector bundle. The orientation homomorphism in singular bordism is defined as follows:
![\displaystyle \rho : \Omega_n^B(X) \to H_n(X; \Zz/2), ~~~ [(M, \bar \nu), f] \mapsto f_*[M].](/images/math/e/6/4/e644fe719a7dc5162365ab6cf4e4b462.png)
If then for all closed
-manifolds
and we can replace the
-coefficients with
-coefficients above.
5 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
-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
![\displaystyle c(M, \bar \nu) : S^{n+r} \to T(E_r)](/images/math/1/3/4/134b576fc0861d734f2939e4191f953e.png)
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
![\displaystyle [c(M, \bar \nu)] =: P((M, \bar \nu)) \in \text{lim}_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).](/images/math/a/9/6/a962b5a6499a87420e136c4030c0aa0f.png)
The celebrated theorem of Pontrjagin and Thom states in part that depends only on the bordism class of
.
Theorem 5.1. There is an isomorphism of abelian groups
![\displaystyle P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto P([M, \bar \nu]).](/images/math/b/5/d/b5dde7798ff8542a4d3599c261e28840.png)
For the proof see [Bröcker&tom Dieck1970, Satz 3.1 and Satz 4.9].
For example, if is the path fibration over
, then
is homotopic to the sphere spectrum
and
is the
-th stable homotopy group. On the other hand, in this case
is the framed bordism group and as special case of Theorem 5.1 we have
Theorem 5.2.
There is an isomorphism .
The Pontrjagin-Thom isomorphism generalises to singular bordism.
Theorem 5.3.
For any space there is an isomorphism of abelian groups
![\displaystyle P : \Omega_n^B(X) \cong \pi_n^S(MB \wedge X_+)](/images/math/5/6/9/56975b2ed1f17d7dce51a5b335c72742.png)
where denotes the smash produce of the specturm
and the space
with a disjoint basepoint added.
6 Spectral sequences
For any generalised homology theory there is a spectral sequence, called the Atiyah-Hirzebruch spectral sequence (AHSS) which can be used to compute
. The
term of the AHSS is
and one writes
![\displaystyle \bigoplus_{p+q = n} H_p(X; h_q(\text{pt})) \Longrightarrow h_{n}(X).](/images/math/6/1/3/6130b6eceed985789d71d42ee60ea81e.png)
The Pontrjagin-Thom isomorphisms above therefore give rise to the following theorems. For the first we recall that stable homotopy defines a generalised homology theory, and we use the Thom isomorphism with local coefficients: .
Theorem 6.1.
Let be a fibred stable vector bundle. There is a spectral sequence
![\displaystyle \bigoplus_{p+q = n} H_p(B;\underline{\pi_q^S}) \Longrightarrow \Omega_{n}^B.](/images/math/2/5/f/25f7d62ef907a28eba736f980762b14b.png)
Theorem 6.2.
Let be a fibred stable vector bundle and
a space. There is a spectral sequence
![\displaystyle \bigoplus_{p+q = n} H_p(X; \Omega_q^B) \Longrightarrow \Omega_n^B(X).](/images/math/9/6/f/96f7da4561aa1b6c1951c366965f613a.png)
Next recall Serre's theorem [Serre1951] that vanishes unless
in which case
. From the above spectral sequences of Theorems 6.1 and 6.2 we deduce the following
Theorem 6.3 Cf. [Kreck&Lück2005, Thm 2.1].
If then the orientation homomorphism induces an isomorphism
![\displaystyle \rho \otimes \id_{\Qq} : \Omega_n^B \otimes \Qq \cong H_n(B; \Qq).](/images/math/a/e/2/ae22cca96d1bb9c77598e1564353ca33.png)
Moreover for any space ,
and if
is connected, the rationalised orientation homomorphism
may be identified with the projection
![\displaystyle \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq)) \to H_n(B; H_0(B; \Qq)) = H_n(B; \Qq).](/images/math/2/b/6/2b682819fbac72832ff540ad4ba53555.png)
7 Piecewise linear and topological bordism
Let and
denote respectively the classifying spaces for stable piecewise linear homeomorphisms of Euclidean space and origin-preserving homeomorphisms of Euclidean space. Note that while there are honest groups
and
, the piecewise linear case requires more care.
If or
, and
is a fibration, and
is a compact
manifold then just as above, we can define an
-structure on
to be an equivalence class of lifts of of the classifying map of the stable normal bundle of
:
![\displaystyle \xymatrix{ & B \ar[d]^{\gamma} \\ M \ar[r]_{\nu_M} \ar[ur]^{\bar \nu} & BCAT.}](/images/math/b/a/5/ba51fdb8a9344b550c87b0eac357a68f.png)
Note that manifolds have stable normal
bundles classified by
.
Just as before we obtain bordism groups of closed n-dimensional
-manifolds with
structure
![\displaystyle \Omega_n^B : = \{ [M, \bar \nu ]\}.](/images/math/2/a/f/2af6e1e7c515cfbc5e15df8f55f9cf1a.png)
The fibration again defines a Thom spectrum
and one asks if there is a Pontrjagin-Thom isomorphism. The proof of the Pontrjagin-Thom theorem relies on transversality for manifolds and while this is comparatively easy in the
-category, it is was a major breakthrough to achieve this for topological manifolds: achieved in [Kirby&Siebenmann1977] for dimensions other than 4 and then in [Freedman&Quinn1990] in dimension 4. Thus one has
Theorem 7.1.
There is an isomorphism .
The basic bordism groups for and
manifolds,
and
, are denoted by
,
,
and
. Their computation is significantly more difficult than the corresponding bordism groups of smooth manifolds: there is no analogue of Bott periodicity for
and
and so the spectra
and
are far more complicated. For now we simply refer the reader to [Madsen&Milgram1979, Chapters 5 & 14] and [Brumfiel&Madsen&Milgram1973].
![O \to PL](/images/math/1/0/c/10cc50b448e7ae086b08250b14dcf547.png)
![O \to TOP](/images/math/5/a/1/5a1b14b5d26b24ba083e2d462ecc43e8.png)
![\displaystyle \pi_i(O) \otimes \Qq \cong \pi_i(PL) \otimes \Qq ~~ \text{and} ~~ \pi_i(O) \otimes \Qq \cong \pi_i(TOP) \otimes \Qq ~~\forall i.](/images/math/d/1/0/d109978aee91bde721cb83533c7bf183.png)
As a consequence one has
Theorem 7.2. There are isomorphisms
![\displaystyle \Omega_i^{TOP} \otimes \Qq = \Omega_i^{PL} \otimes \Qq = 0 ~~ \forall i,](/images/math/e/3/7/e374b652b948b7a70cab53ab4566bbe1.png)
![\displaystyle \Omega_i^{SO} \otimes \Qq \cong \Omega_i^{SPL} \otimes \Qq \cong \Omega_i^{STOP} \otimes \Qq ~~ \forall i.](/images/math/c/8/a/c8a52e274a5aee17ab85607f02e3dbce.png)
8 References
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