Fusion system and group actions with abelian isotropy subgroups

Abstract

We prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × double struk S signn1 ×... × double struk S signnk for some positive integers n1, ..., nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology. Copyright © Edinburgh Mathematical Society 2013.