Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$

TL;DR

This paper classifies free linear actions of $(Z/p)^2$ on products of spheres, identifying the quotients up to homotopy and homeomorphism using $k$-invariants and Pontrjagin classes, with explicit calculations from rotation numbers.

Contribution

It provides a classification of quotients of sphere products under linear $(Z/p)^2$ actions using algebraic topology tools, including explicit computation methods.

Findings

Classified quotients up to homotopy by $k$-invariants.

Classified quotients up to homeomorphism by Pontrjagin classes.

Developed methods to compute invariants from rotation numbers.

Abstract

For an odd prime $p$, we consider free actions of $(\mathbb{Z}/p)^2$ on $S^{2n-1}\times S^{2n-1}$ given by linear actions of $(\mathbb{Z}/p)^2$ on $\mathbb{R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the $k$-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the $k$-invariants and the Pontrjagin classes from the rotation numbers.