Fusion systems and constructing free actions on products of spheres (Q415468)

If \(G\) is a finite group, then the rank of \(G\) is the largest \(k\) such that \(G\) contains a subgroup isomorphic to \((\mathbb{Z}/p)^k\) where \(p\) is a prime. A theorem of Heller asserts that if \(G\) acts freely on a product of two spheres, then \(rk(G)\leq 2\). Adem, Smith and the first author of this paper showed that if \(p\geq 5\), the every rank two \(p\)-group acts freely and smoothly on a product of two spheres, and Adem and Smith proved that every rank two \(p\)-group acts freely on a CW-complex in the homotopy type of a product of two spheres. The main result in this paper is that a finite \(p\)-group \(G\) acts freely and smoothly on a product of two spheres if and only if \(rk(G)\leq 2\). The authors develop a construction which takes a smooth free action on a manifold and produces a smooth free action on the manifold cross a product of spheres if the isotropy subgroups of the originial action can be associated to a fusion system satisfying certain conditions.