$\mathbb Z_{/p}\times \mathbb Z_{/p}$ actions on $S^n\times S^n$

Abstract

We determine the homotopy type of quotients of $S^n \times S^n$ by free actions of $\mathbb Z_{/p} \times \mathbb Z_{/p}$ where $2p>n+3$. Much like free $\mathbb Z_{/p}$ actions, they can be classified via the first $p$-localized $k$-invariant, but there are restrictions on the possibilities, and these restrictions are sufficient to determine every possibility in the $n=3$ case. We use this to complete the classification of free $\mathbb Z_{/p} \times \mathbb Z_{/p}$ actions on $S^3 \times S^3$, for $p>3$, by reducing the problem to the simultaneous classification of pairs of binary quadratic forms. Although the restrictions are not sufficient to determine which $k$-invariants are realizable in general, they can sometimes be used to rule out free actions by groups that contain $\mathbb Z_{/p}\times\mathbb Z_{/p}$ as a normal Abelian subgroup.