Title: Free Actions of $p$-Groups on Products of Lens Spaces
(To appear in Proceedings of A.M.S.)
Author: Ergun Yalcin (Erg\" un Yal\c c\i n in Latex)
Current Address:
Ergun Yalcin
McMaster University
Department of Mathematics
1280 Main Street West
Hamilton, ON, Canada
L8S 4K1
Abstract: Let $p$ be an odd prime number. We prove that
if $(\ZZ /p)^r$ 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 to 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 $\rk (G)\leq k$ and $G$ has the $\Om$-extension property.
The main technique is to study group extensions associated to free actions.