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 $\dim(M) \leq \dim(N)$ and give a classification of such homomorphisms when $\dim(M) = \dim(N)$.
Categories: math.GT math.DS math.GN math.GR
