Free actions of 𝑝-groups on products of lens spaces

Yalçin, Ergün

doi:10.1090/S0002-9939-00-05756-7

Free actions of $p$-groups on products of lens spaces
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by Ergün Yalçin

Proc. Amer. Math. Soc. 129 (2001), 887-898

DOI: https://doi.org/10.1090/S0002-9939-00-05756-7

Published electronically: September 20, 2000

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Abstract:

Let $p$ be an odd prime number. We prove that if $(\mathbf {Z}/p)^r$ acts freely on a product of $k$ equidimensional lens spaces, then $r\leq k$. This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian $p$-groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A $p$-group can act freely on a product of $k$ lens spaces with a trivial action on homology if and only if $\operatorname {rk}(G)\leq k$ and $G$ has the $\Omega$-extension property. The main technique is to study group extensions associated to free actions.

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