Free actions of finite groups on products of Dold manifolds

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 $\prod _{i=1}^{k}P(2m_i,n_i)$, then $G\cong ({\mathbb{Z}}_2{)}^l$ for some $l\le k$. This is achieved by first proving a similar assertion for $\prod _{i=1}^k{S}^{2m_i}\times \mathbb{C}{P}^{n_i}$. 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.

17 pages, Theorems 1.3 and 1.4 have been improved, introduction modified