Structure theorems for actions of homeomorphism groups

2019/02/13 by Chen, Lei, Mann, Kathryn · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1902.05117

Abstract

We give general classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys' "extension problem" for diffeomorphisms of manifolds with boundary, as well as a classification of all homomorphisma Homeo₀(M) → Homeo₀(N) when dim(M) = dim(N) (and related results for diffeomorphisms), and a complete classification of actions of Homeo₀(S¹) on surfaces. This resolves many problems in a program initiated by Ghys, and gives definitive answers to conjectures of Militon and Hurtado and a question of Rubin.

Cited by

• The Euler Class and Flux Homomorphisms under Non-Orientability

Related

• Groups of homeomorphisms of one-manifolds, I: actions of nonlinear groups
• Actions of groups of homeomorphisms on one-manifolds
• Actions of homeomorphism groups of manifolds admitting a nontrivial finite free action
• Actions of groups of diffeomorphisms on one-manifolds
• There are no exotic actions of diffeomorphism groups on 1-manifolds