#### Volume 13, issue 1 (2009)

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Global fixed points for centralizers and Morita's Theorem

### John Franks and Michael Handel

Geometry & Topology 13 (2009) 87–98
##### 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 $3$.

##### Keywords
mapping class group, pseudo-Anosov, global fixed point, lifting problem
##### Mathematical Subject Classification 2000
Primary: 37E30, 57M60, 37C25