Isotopy versus equivariant isotopy in dimensions 3 and higher

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 that 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.