Free actions of finite groups on products of Dold manifolds

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 $l\le k$. 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.