Free actions of finite groups on products of Dold manifolds

TL;DR

This paper studies free actions of finite groups on products of Dold manifolds, showing such groups are elementary 2-groups and characterizing the cohomology of orbit spaces, with implications for equivariant maps.

Contribution

It establishes that finite groups acting freely and mod 2 cohomologically trivially on products of Dold manifolds are elementary 2-groups, and describes the cohomology of their orbit spaces.

Findings

Finite group actions are isomorphic to (Z_2)^l.

Characterization of mod 2 cohomology algebra of orbit spaces.

Application to Z_2-equivariant maps.

Abstract

The Dold manifold $ P(m,n)$ is the quotient of $S^m \times \mathbb{C}P^n$ by the free involution that acts antipodally on $ S^m $ and by complex conjugation on $ \mathbb{C}P^n $. In this paper, we investigate free actions of finite groups on products of Dold manifolds. We show that if a finite group $ G $ acts freely and mod 2 cohomologically trivially on a finite-dimensional CW-complex homotopy equivalent to ${\displaystyle \prod_{i=1}^{k} P(2m_i,n_i)}$, then $G\cong (\mathbb{Z}_2)^l$ for some $l\leq k$. This is achieved by first proving a similar assertion for $ \displaystyle \prod_{i=1}^{k} S^{2m_i} \times \mathbb{C} P^{n_i} $. We also determine the possible mod 2 cohomology algebra of orbit spaces of arbitrary free involutions on Dold manifolds, and give an application to $ \mathbb{Z}_2 $-equivariant maps.