**"**** Group Cohomology
and Control of p-Fusion****"**

By

** Abstract:** I will report on recent
joint work with Dave Benson and Ellen Henke. We show that if an inclusion of
finite groups H < G of index prime to p induces an F-isomorphism of mod p cohomology rings, or equivalently a homeomorphism of cohomology varieties, then H controls p-fusion in G, if p is
odd -- in particular the map on cohomology is in fact
an isomorphism. This generalizes classical results of Quillen,
who proved this when H is the Sylow p-subgroup, and
also 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, and using Hopkins-Kuhn-Ravenel character
theory. The results follow from of a general algebraic theorem we prove about
p-fusion systems, that roughly says that isomorphisms
between p-fusion systems are detected on elementary abelian
p-groups. My talk will be a gentle introduction to these results, and the
surrounding mathematics.

**Date: ****Wednesday, November 27, 2013**

**Time: ****15.40 – 16:30 **

**Place: ****Mathematics Seminar Room,
SA-141**

Tea and cookies will be served after the
seminar.