Cohomology algebra of orbit spaces of free involutions on lens spaces

TL;DR

This paper classifies the mod 2 cohomology algebras of orbit spaces resulting from free involutions on lens spaces, providing a comprehensive understanding of their topological structure and applications to equivariant maps.

Contribution

It offers a complete classification of cohomology algebras of orbit spaces under free involutions on lens spaces, including explicit examples and applications.

Findings

Classified possible mod 2 cohomology algebras of orbit spaces

Provided explicit examples of spaces with these cohomology algebras

Applied results to non-existence of certain equivariant maps

Abstract

Let $G$ be a group acting continuously on a space $X$ and let $X/G$ be its orbit space. Determining the topological or cohomological type of the orbit space $X/G$ is a classical problem in the theory of transformation groups. In this paper, we consider this problem for cohomology lens spaces. Let $X$ be a finitistic space having the mod 2 cohomology algebra of the lens space $L_p^{2m-1}(q_1,...,q_m)$. Then we classify completely the possible mod 2 cohomology algebra of orbit spaces of arbitrary free involutions on $X$. We also give examples of spaces realizing the possible cohomology algebras. In the end, we give an application of our results to non-existence of $\mathbb{Z}_2$-equivariant map $\mathbb{S}^n \to X$.