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 Lemma 2.1 of Connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
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
if ( does not admit a -structure) then it's normal -type is the fibration
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].
There are versions of the concepts and results of this page for -manifolds or topological manifolds. Everything is the same if one replaces the normal bundles by the or topological normal bundles and by the corresponding classifying spaces an . Then one replaces the smooth -bordism groups by the corresponding or topological -bordism groups B-Bordism. Then Theorem 3.1. holds for or topological manifolds where we classify up to stable -homeomorphisms or stable homeomorphisms.
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