Continuity of discrete homomorphisms of diffeomorphism groups

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 $dim\left(M\right)=dim\left(N\right)$.