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L2 - Invariants

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Bryan Clair, SLU

  • Topology Seminar
When Thu, Sep 07, 2000
from 12:00 PM to 01:00 PM
Where RH 142
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Abstract: Let X be a compact manifold or cell complex, and \tilde{X} be an infinite covering space with group of deck transformations G. By studying L2-forms (or cochains) on \tilde{X}, one can define topological invariants of X. The most classical of these are the L2 Betti numbers, first considered by Atiyah in 1976. Like the ordinary Betti numbers of a compact space, the L2 Betti numbers are the dimension of a homology group. However, the homology groups coming from L2 -forms are infinite dimensional Hilbert spaces, so the machinery of von Neumann algebras is needed to provide a quantitative dimension. I will spend the first lecture on background material, and basic examples, with some older theorems and big conjectures thrown in for spice. Don't worry if you don't know anything about von Neumann algebras. I plan to talk about Gromov's result that the L2 Betti numbers vanish when X is aspherical and G is amenable. I'll prove Luck's theorem, which says that ordinary Betti numbers (renormalized) of a tower of covering spaces converge to the L2 Betti numbers of the limit covering. I will also discuss more refined L2 invariants such as the Novikov-Shubin invariants and L2 -Reidemeister torsion.

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