Uniform Lefschetz fixed-point theory

Daisuke Kishimoto (Kyushu University)

Monday 23rd March 15:00-16:00
Maths 311B

Abstract

I will talk about joint work with Tsuyoshi Kato and Mitsunobu Tsutaya on the Lefschetz fixed-point theory for noncompact manifolds. We define the Lefschetz class of a uniformly continuous self-map of a noncompact manifold of bounded geometry, which stays within a bounded distance from the identity map, as an element of Block and Weinberger’s uniformly finite homology. We then prove that the Lefschetz class is zero if and only if the map is uniformly homotopic to a strongly fixed-point free map. To achieve this, we introduce a new cohomology for metric spaces, called uniform bounded cohomology, and develop an obstruction theory based on it.

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