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
Publication
Received: 23 April 2008
Revised: 9 September 2008
Accepted: 26 July 2008
Preview posted: 22 October 2008
Published: 1 January 2009
Proposed: Benson Farb
Seconded: Leonid Polterovich, Shigeyuki Morita
Authors
 John Franks Department of Mathematics Northwestern University Evanston, IL 60208 Michael Handel Department of Mathematics Lehman College Bronx, NY 10468