Free actions on products of real projective spaces

TL;DR

This paper establishes an upper bound on the rank of elementary abelian 2-groups acting freely on products of real projective spaces, extending a conjecture of Cusick through homotopy-theoretic methods.

Contribution

It proves a new bound on free actions of $( ext{Z}/2)^r$ on products of real projective spaces, generalizing previous conjectures with a homotopy-theoretic approach.

Findings

Bound on the rank of free $( ext{Z}/2)^r$ actions is given.

The result confirms a homotopy-theoretic version of Cusick's conjecture.

Provides a new perspective on group actions on topological spaces.

Abstract

We prove that if $G=(\mathbb{Z}/2)^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\mathbb{R}P ^{n_1} \times \cdots \times \mathbb{R} P ^{n_k}$ with trivial action on the mod-$2$ cohomology, then $r \leq \mu (n_1)+ \cdots + \mu(n_k )$ where for each integer $n\geq 0$, $\mu (n)=0$ if $n$ is even, $\mu(n)=1$ if $n\equiv 1$ mod 4, and $\mu(n)=2$ if $n\equiv 3$ mod 4. This proves a homotopy-theoretic version of a conjecture of Cusick.