Continuity of discrete homomorphisms of diffeomorphism groups

TL;DR

This paper proves that any group homomorphism between diffeomorphism groups of closed manifolds is continuous, and classifies such homomorphisms when the manifolds have the same dimension, revealing structural constraints.

Contribution

It establishes the continuity of discrete homomorphisms between diffeomorphism groups and provides a classification in the equal-dimension case.

Findings

Any group homomorphism between $ ext{Diff}_c(M)$ and $ ext{Diff}_c(N)$ is continuous.

A non-trivial homomorphism implies $ ext{dim}(M) \u2264 ext{dim}(N)$.

Classification of homomorphisms when $ ext{dim}(M) = ext{dim}(N)$.

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 $\dim(M) \leq \dim(N)$ and give a classification of such homomorphisms when $\dim(M) = \dim(N)$.