Global fixed points for centralizers and Morita’s Theorem

Franks, John; Handel, Michael

doi:10.2140/gt.2009.13.87

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Global fixed points for centralizers and Morita's Theorem

John Franks and Michael Handel

Geometry & Topology 13 (2009) 87–98

DOI: 10.2140/gt.2009.13.87

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

References

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

Volume 13, issue 1 (2009)