Continuity of discrete homomorphisms of diffeomorphism groups

Abstract

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

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Cited by in corpus (8)

• Big mapping class groups: an overview
• Automatic continuity for homeomorphism groups and applications
• Structure theorems for actions of homeomorphism groups
• Diffeomorphism groups of critical regularity
• The Burnside problem for ${\text{Diff}}_{\text{Vol}}({\mathbb{S}}^2)$
• Braid groups and discrete diffeomorphisms of the punctured disk
• PL(M) admits no Polish group topology
• Homomorphisms between pure mapping class groups