Orbit spaces of free involutions on the product of two projective spaces

TL;DR

This paper investigates free involutions on spaces with cohomology like a product of two projective spaces, determining their orbit space cohomology and implications for equivariant maps.

Contribution

It characterizes the mod 2 cohomology algebra of orbit spaces under free involutions on such spaces using spectral sequences.

Findings

Classifies possible cohomology algebras of orbit spaces.

Shows non-existence of certain equivariant maps for these spaces.

Provides tools for studying involutions on product spaces.

Abstract

Let $X$ be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on $X$ and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration $X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}$. We also give an application of our result to show that if $X$ has the mod 2 cohomology algebra of the product of two real projective spaces (respectively complex projective spaces), then there does not exist any $\mathbb{Z}_2$-equivariant map from $\mathbb{S}^k \to X$ for $k \geq 2$ (respectively $k \geq 3$), where $\mathbb{S}^k$ is equipped with the antipodal involution.