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Free actions of $p$-groups on products of lens spaces

Author: Ergün Yalçin
Journal: Proc. Amer. Math. Soc. 129 (2001), 887-898
MSC (2000): Primary 57S25; Secondary 20J06, 20D15
Published electronically: September 20, 2000
MathSciNet review: 1792188
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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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Additional Information

Ergün Yalçin
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Address at time of publication: Department of Mathematics, Bilkent University, Ankara, Turkey 06533

Keywords: Group actions, products of lens spaces, group extensions
Received by editor(s): May 12, 1999
Published electronically: September 20, 2000
Communicated by: Ralph Cohen
Article copyright: © Copyright 2000 American Mathematical Society

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