Fixity and free group actions on products of spheres

Abstract

A representation G ⊂ U(n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U(n) /U(n-f-1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on $\mathbb S^{2n-1}\times\mathbb S^{4n-5}.$ We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain (ℤ /p)³ as a subgroup.

We use propagation methods from surgery theory to show that a representation of fixity f < n - 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n - 1)!. We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described.