Continuity of discrete homomorphisms of diffeomorphism groups

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 $Φ: \text{Diff}_c(M) \to \text{Diff}_c(N)$ implies that $\dim(M) \leq \dim(N)$ and give a classification of such homomorphisms when $\dim(M) = \dim(N)$.

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