## 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 ${\displaystyle \prod_{i=1}^{k} P(2m_i,n_i)}$, 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.Comment: 17 pages, Theorems 1.3 and 1.4 have been improved, introduction modifie

Topics: Mathematics - Algebraic Topology, 57S25 (Primary) 57S17, 55T10 (Secondary)
Year: 2019
OAI identifier: oai:arXiv.org:1809.02307

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