Global fixed points for centralizers and Morita's Theorem

Abstract

We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeomorphism with infinitely many global fixed points. As another application we give an elementary proof of Morita's Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of diffeormorphisms of $S$ and we improve the lower bound for $g$ from 5 to 3.

References in corpus (2)

• Low-dimensional homology groups of mapping class groups: a survey
• Fixed Points of abelian actions on $S^2$

Cited by in corpus (8)

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• Cohomological obstructions to Nielsen realization
• A periodicity criterion and the section problem on the Mapping Class Group
• Non-realizability of the Torelli group as area-preserving homeomorphisms