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

TL;DR

This paper classifies free actions of the group /p /p on products of spheres, determining the homotopy types of quotients and completing classifications for specific cases, using algebraic invariants and quadratic forms.

Contribution

It provides a complete classification of free /p /p actions on S^3  S^3 for p>3, extending previous results and analyzing restrictions on k-invariants.

Findings

Classified homotopy types of quotients for certain group actions.

Completed classification for /p /p actions on S^3  S^3 when p>3.

Identified restrictions on k-invariants and their implications.

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.