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 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 $C^2$ diffeomorphisms of $S$ and we improve the lower bound for $g$ from $5$ to $3$.