Isotopy versus equivariant isotopy in dimensions three and higher

TL;DR

This paper investigates the difference between isotopy and equivariant isotopy for group actions on manifolds in higher dimensions, providing criteria and examples where they differ, and exploring implications for automorphism groups and fibered manifolds.

Contribution

It establishes a general criterion for when isotopic equivariant diffeomorphisms are not equivariantly isotopic in higher dimensions, with explicit examples and applications.

Findings

Counterexamples to equivariant isotopy equivalence in higher dimensions

Construction of an invariant using homology of an infinite cover

Applications to automorphism groups and fibered manifolds

Abstract

Given a finite group action on a smooth manifold, we study the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? Birman-Hilden and Maclachlan-Harvey proved the answer is "yes" for most surfaces. By contrast, we give a general criterion in higher dimensions under which there are many equivariant diffeomorphisms which are isotopic but not equivariantly isotopic. Examples satisfying this criterion include branched covers of split links and "stabilized" branched covers. We prove the result by constructing an invariant valued in the homology of a certain infinite cover of the manifold. We give applications to outer automorphism groups of free products and to group actions on manifolds which fiber over the circle.