Torsion, L^2 cohomology and complexity

Abstract: Atiyah introduced real valued L^2 betti numbers as a way of understanding the (usually infinite dimensional) cohomology of universal covers of finite complexes. As far as anyone knows these are always integers for torsion free fundamental group, but for groups with torsion very much more exotic possibilities arise.

We will use this and an invariant of Cheeger and Gromov to see that whenever an oriented smooth manifold of dimension 4k+3 has torsion in its fundamental group, there are many other manifolds homotopy equivalent but not diffeomorphic to it and that in the known situations where betti numbers can be irrational there is even an infinitely generated group of such! And, I will also use this invariant to explain how many simplices (roughly) it takes to build a standard Lens space. This is based on old work with Stanley Chang, and recent work with Geunho Lim.

differential geometrydynamical systemsgeometric topologysymplectic geometryspectral theory

Audience: researchers in the topic

Geometry and Dynamics seminar

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