Volume 19, issue 4 (2015)

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Continuity of discrete homomorphisms of diffeomorphism groups

Geometry & Topology 19 (2015) 2117–2154
Abstract

Let $M$ and $N$ be two closed ${C}^{\infty }$ manifolds and let ${Diff}_{c}\left(M\right)$ denote the group of ${C}^{\infty }$ diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between ${Diff}_{c}\left(M\right)$ and ${Diff}_{c}\left(N\right)$ is continuous. We also show that a nontrivial group homomorphism $\Phi :{Diff}_{c}\left(M\right)\to {Diff}_{c}\left(N\right)$ implies that $dim\left(M\right)\le dim\left(N\right)$ and give a classification of such homomorphisms when $dim\left(M\right)=dim\left(N\right)$.

Keywords
example, sample layout
Mathematical Subject Classification 2010
Primary: 57M99
Secondary: 55Q33, 55Q32
Publication
Received: 13 September 2013
Revised: 4 August 2014
Accepted: 19 September 2014
Published: 29 July 2015
Proposed: Danny Calegari
Seconded: Benson Farb, Leonid Polterovich
Authors
 Sebastian Hurtado Department of Mathematics University of California Berkeley, CA 94720 USA