Stable classification of manifolds
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1 Introduction
Two closed smooth manifolds and
of dimension
are called stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies of
. Note that since
has an orientation reversing diffeomorphism the connect sum with it is well defined (see Parametric connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
![\nu_M: M \to BO](/images/math/7/0/2/702a99eb9c03176eaf7836e2462d66e3.png)
![r](/images/math/a/a/7/aa7bb4b05fbd27db7ca594893f166b47.png)
![\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO](/images/math/3/e/8/3e8676b9f3f95d07a11cead9a623b252.png)
![\nu_M](/images/math/1/0/b/10b3daff6af9719574f9834d9f7da15e.png)
![p_r: B_r \to BO](/images/math/3/a/2/3a25ab86418fa3ea9367ba994b978074.png)
![B_r](/images/math/1/0/3/10389f84752eca3af2ad944cc136761f.png)
![CW](/images/math/4/d/3/4d3c0917a30c46d2cbc6801f69790c9b.png)
![p_r](/images/math/4/0/d/40d1d7ce1aaead8d4a45d077cd1b9cb6.png)
![r](/images/math/a/a/7/aa7bb4b05fbd27db7ca594893f166b47.png)
![\ge r](/images/math/8/8/2/8824d519ea32dadfd6d590a4dabb5fd9.png)
![\bar \nu_r](/images/math/7/a/c/7ac58c8c0d57eeffbab1afd85c14d6de.png)
![r](/images/math/a/a/7/aa7bb4b05fbd27db7ca594893f166b47.png)
Definition 1.1. The fibre homotopy type of the fibration is an invariant of
and is called the normal
-type of
denoted
[Kreck1999].
In particular, the normal -type does not depend on the choice of an embedding.
For example the normal -type of an oriented manifold is the universal covering
![\displaystyle BSO \to BO,](/images/math/a/d/4/ad428d6be57742ae92f0f3ea6cb1e72d.png)
![0](/images/math/d/0/a/d0a87271a40bebf0cd626354a0c0aee2.png)
![\displaystyle BO \to BO.](/images/math/d/8/3/d8382f0c174490f61e46f8822db9b4af.png)
![1](/images/math/0/6/d/06d3730efc83058f497d3d44f2f364e3.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![w_2(M) =0](/images/math/f/e/0/fe06d4c2f7c8f0ffc95102b123c99ba3.png)
![Spin](/images/math/6/e/1/6e1c4c882f5c1badb0d9850afd5c40e8.png)
![1](/images/math/0/6/d/06d3730efc83058f497d3d44f2f364e3.png)
![\displaystyle BSpin \to BSO,](/images/math/8/8/f/88f53a01fa8de3252e5d93a94e290af8.png)
if (
does not admit a
-structure) then it's normal
-type is the fibration
![\displaystyle BSO \to BO.](/images/math/e/8/6/e8615cb39dff474e894722604dd245e3.png)
![1](/images/math/0/6/d/06d3730efc83058f497d3d44f2f364e3.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,](/images/math/1/0/9/1094e2876b36b1c21de2a3b8fd2186cf.png)
![p](/images/math/2/a/0/2a039ed8fdbf4ceaa9e79cdc3aecd1a2.png)
![BSpin](/images/math/d/f/8/df871d85bd087a8d4480af885e3e1dad.png)
![BSpin \to BO](/images/math/7/2/5/725e9595f8ba8b0b7d393d89f84fdf11.png)
![1](/images/math/0/6/d/06d3730efc83058f497d3d44f2f364e3.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\tilde M](/images/math/4/e/8/4e86f23e55ff5456513c266fb250a6f9.png)
![Spin](/images/math/6/e/1/6e1c4c882f5c1badb0d9850afd5c40e8.png)
![\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,](/images/math/9/0/e/90e7b641295474ee96af9b6cc799f123.png)
![p](/images/math/2/a/0/2a039ed8fdbf4ceaa9e79cdc3aecd1a2.png)
![BSO](/images/math/0/c/d/0cd019226eac79674d9c85982e4b9696.png)
![BSO \to BO](/images/math/2/4/7/247cc8711816b92898df0b00184fb1ea.png)
![\tilde M](/images/math/4/e/8/4e86f23e55ff5456513c266fb250a6f9.png)
![Spin](/images/math/6/e/1/6e1c4c882f5c1badb0d9850afd5c40e8.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
Definition 1.2. If is the normal
-type of a manifold
, the choice of a homotopy class of lifts
, which is a
-equivalence is called a normal
-smoothing. We say that two normal
-smoothings
and
with same normal
-type are diffeomorphic, if there is a diffeomorphism
compatible with the normal
-smoothings. The latter means that if we have embedded
into
, we embed
via the composition with
, so that
. Then we require that
id fibre homotopic to
.
The group of homotopy classes of fibre homotopy self equivalences acts by composition on the normal
-smoothings and this action is free and transitive [Kreck1999]. Thus if one fixes a normal
-smoothing the composition with elements of
is a bijection from
to the different normal
-smoothings.
The stable classification of normal (k-1)-smoothings on 2k-dimensional manifolds
Above we defined stable diffeomorphisms. We can define stable diffeomorphisms of normal -smoothings of
-dimensional manifolds
as follows. Suppose that we have two
-dimensional Manifolds
and
with same normal
-type
. We consider
as the boundary of
and equip it with the restriction of the the unique (up to homotopy) lift of the stable normal bundle to
. Using this we obtain a lift of the stable normal bundle of
to
and consider its restriction to
. If
is a normal
-smoothing of
this together with the constructed lift of the stable normal bundle of
to
induces a well defined normal
-smoothing of
which we call
. We say that two normal
-smoothings in
are stably diffeomorphic if the stabilized normal
-smoothings are diffeomorphic.
Theorem 3.1. [Kreck1999] Let and
be
-dimensional closed smooth manifolds with same normal
-type
. Then two normal
smoothings
and
are stably diffeomorphic if and only if the bordism classes of
and
agree in the
-bordism group
and the Euler characteristics agree:
.
If and
are compact manifolds with boundary and
is a diffeomorphism compatible wirth the restriction of the normal
-smoothings to the boundaries, then
extends to a stable diffeomorphism of the normal
-smoothings if and only if
and the closed manifold obtained by gluing
to
via
together with the normal structure given by
and
is zero bordant in
. Here by
we mean
equipped with the normal
-smoothing which is given by the restriction to
of the obvious normal structure on
extending the given structure on
.
Thus two -dimensional closed manifolds
and
are stably diffeomorphic if and only if
, they have the same normal
-type
and admit bordant normal
-smoothings in
.
For simply connected closed smooth -manifolds Wall [Wall1964] showed that they are stably diffeomorphic if and only both admit a
-structure or both don't admit a
-structure, the signatures and the Euler characteristics agree (he uses a different formulation in terms of the intersection forms but by the stable classification of unimodular quadratic forms this is equivalent to our conditions). This is a consequence of the Theorem 3.1. Namely under Wall's condition the normal
-types agree and the
-bordisms groups correspond to
bordism resp. oriented bordism groups which are detected by the signature. Thus we obtain
Corollary 3.2. [Wall1964] Let and
be simply connected closed smooth
-manifolds, then
and
are stably diffeomorphic if and only if they are both spinnable or both not spinnable and the signature and Euler characteristics agree.
Wall proves his theorem in two steps, the first is an (unstable) classification of simply connected -manifolds up top
-cobordism. The second is the proof of a stable
-cobordism theorem in dimension
. The first part is a very special application of a big theory: Surgery. The proof of Theorem 3.1 is a special case of a modified surgery theory [Kreck1999].
An unstable classification of some normal (k-1)-smoothings on 2k-dimensional manifolds
If the normal
-type is the same if and only if the surfaces are both orientable or both non-orientable and one has a better Theorem, namely an unstable classification by the Euler characteristic (in this case the bordism class is determined by the Euler characteristic mod
). It is an interesting question, under which conditions one gets an unstable classification in higher dimensions. For manifolds with finite fundamental group one has the following result for
:
Theorem 6.1. [Kreck1999] Let and
be
-dimensional compact smooth manifolds with same normal
-type
and
. Suppose that if the fundamental group is trivial and, if
is even,
is of the form
, or if the fundamental group is finite and
is of the form
.
Then two normal smoothings
and
are diffeomorphic extending a diffeomorphism of the boundaries compatible with the normal
-smoothings, if and only if
e(N) and the bordism classes of
and
the closed manifold obtained by gluing
to
via
together with the normal structure given by
and
is zero bordant in
.
References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Spanier1981] E. H. Spanier, Algebraic topology, Springer-Verlag, 1981. MR666554 (83i:55001) Zbl 0810.55001
- [Wall1964] C. T. C. Wall, Diffeomorphisms of
-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101