|GÖKOVA GEOMETRY / TOPOLOGY PROCEEDINGS|
||Proceedings of Gökova Geometry-Topology Conference 2015|
||Thoughts about a good classification of manifolds|
"Good" is a matter of taste. But there are mathematical concepts which most mathematicians agree on that they are good. For example if one wants to classify complex vector bundles it seems to be a good idea to look at isomorphism classes of vector bundles modulo addition of trivial vector bundles. The resulting set of equivalence classes is denoted by
, and for compact spaces this is a group under the operation of Whitney sum. Another reason why it is good, is that
is the degree 0 subgroup of a generalized cohomology theory which allows an attack by the standard tools of algebraic topology like exact sequences or spectral sequences.
In this note we take this as a model for the classification of closed connected manifolds. In analogy to the vector bundles we consider diffeomorphism classes of smooth manifolds modulo connected sum with a "trivial" manifold T. Whereas we don't see a good candidate for T for odd dimensional manifolds, we take T = Sn×Sn in even dimensions and pass to what we call the reduced stable diffeomorphism classes of manifolds. In contrast to vector bundles the reduced stable diffeomorphism classes of smooth manifolds don't form a group. But we will see that they decompose as quotients of groups by a linear action of another group. Most of the results in this note are not new, they are all based on the results of my papers , . But we add a perspective which readers might find good.
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Last updated: October 2016
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