Title: Fusion systems and constructing free actions on 
products of spheres

Authors: {\" O}zg{\" u}n {\" U}nl{\" u} and Erg{\" u}n Yal{\c c}{\i}

Status: To appear in Math. Zeit.

Address: 
Department of Mathematics, Bilkent University,
06800 Bilkent, Ankara, Turkey



Abstract:
We show that every rank two $p$-group acts freely and
 smoothly 
on a product of two spheres.  This follows from a more
 general 
construction: given a smooth action of a finite group $G$ on
a 
manifold $M$, we construct a smooth free action on 
$M \times \bbS
^{n_1} \times \dots \times \bbS ^{n_k}$ when 
the set of isotropy 
subgroups of the $G$-action on $M$ can be 
associated to a fusion 
system satisfying certain properties. 
Another consequence of this
 construction is that if $G$ is an 
(almost) extra-special $p$-group 
of rank $r$, then it acts freely 
and smoothly on a product of $r$ 
spheres.