Continuity of discrete homomorphisms of diffeomorphism groups
Sebastian Hurtado

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 and be two closed manifolds and let denote the group of diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between and is continuous. We also show that a non-trivial group homomorphism implies that and give a classification of such homomorphisms when .
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