\title[Variety isomorphism and control of $p$--fusion]{
Variety isomorphism in group cohomology and control of
$p$--fusion}
\author[D.~Benson]{David Benson}
\address{Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE}
\author[J.~Grodal]{Jesper Grodal}
\address{Department of Mathematical Sciences,  University of Copenhagen, Universitetsparken 5, 
DK--2100 Copenhagen, Denmark}
\author[E.~Henke]{Ellen Henke}
\address{Department of Mathematical Sciences, 
University of Copenhagen, Universitetsparken 5, 
DK--2100 Copenhagen, Denmark}
\thanks{All three authors were supported by the Danish National Research
  Foundation (DNRF) through the Centre for Symmetry and
  Deformation. The second author was also supported by an ESF EURYI grant.}

\begin{document}

\begin{abstract}

We show that if an inclusion of finite groups $H \leq G$ of index
prime to $p$ induces a homeomorphism of mod $p$ cohomology varieties, 
or equivalently an $F$--isomorphism in mod
$p$ cohomology, then $H$ controls $p$--fusion
in $G$, if $p$ is odd. This generalizes classical results of Quillen
who proved this when $H$ is a Sylow $p$-subgroup, and furthermore
implies a hitherto difficult result of Mislin about cohomology
isomorphisms. For $p=2$ we give analogous
results, at the cost of replacing mod $p$ cohomology with higher
chromatic cohomology theories.

The results are consequences of a general algebraic
theorem we prove, that says that
isomorphisms between $p$--fusion systems over the same finite $p$--group
are detected on ele\-mentary
abelian $p$--groups if $p$ odd and abelian $2$--groups of exponent at most
$4$ if $p=2$.
\end{abstract}