Actions of groups of diffeomorphisms on one-manifolds

TL;DR

This paper proves that for certain closed manifolds fibering over spheres, any group homomorphism from their diffeomorphism groups to the homeomorphism group of the real line is trivial, revealing rigidity in their group actions.

Contribution

It establishes the triviality of homomorphisms from diffeomorphism groups of specific manifolds to the homeomorphism group of the line, extending understanding of group action rigidity.

Findings

Homomorphisms from $ ext{Diff}_c(M)_0$ to $ ext{Homeo}( eal)$ are trivial for manifolds fibering over spheres.

Demonstrates rigidity of diffeomorphism groups acting on the real line.

Provides new insights into the structure of diffeomorphism groups of fibered manifolds.

Abstract

Denote by $\DC(M)_0$ the identity component of the group of compactly supported $C^\infty$ diffeomorphisms of a connected $C^\infty$ manifold $M$, and by $\HR$ the group of the homeomorphisms of $\R$. We show that if $M$ is a closed manifold which fibers over $S^m$ ($m\geq 2$), then any homomorphism from $\DC(M)_0$ to $\HR$ is trivial.