Stable classification of manifolds
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Two closed smooth manifolds $M$ and $N$ of dimension $2n$ are called '''stably diffeomorphic''' if there is an integer $r$ such that $M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$ we mean the connected sum with $r$ copies of $S^n \times S^n$. Note that since $S^n \times S^n$ 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. | Two closed smooth manifolds $M$ and $N$ of dimension $2n$ are called '''stably diffeomorphic''' if there is an integer $r$ such that $M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$ we mean the connected sum with $r$ copies of $S^n \times S^n$. Note that since $S^n \times S^n$ 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. |
Revision as of 15:32, 26 November 2010
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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
of [Spanier1981], page 440. This means that is a fibration, where is a -complex, the map is -connected, i.e. the homotopy groups of the fibre vanish in degree , and the map is an -equivalence.
Definition 1.1. The fibre homotopy type of the fibration is an invariant of the map and is called the normal -type of denoted [Kreck1999].
For example the normal -type of an oriented manifold is the universal covering
whereas the normal -type of a non-orientable manifold is the identity map
The normal -type of a simply connected manifold is the fibration
,if admits a -structure (if and only if the Stiefel-Whintey class vanishes) and the fibration
if does not admit a structure. More generally, the normal -type of a Spin-manifold is
where is the composition of the projection to and the projection . The normal -type of a manifold such that the universal covering does not admit a -structure, is
where is the composition of the projection to and the projection . The case where admits a -structure but doesn't, is treated in Stable classification of 4-manifolds.
If
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