Group cohomology and control of p–fusion

We show that if an inclusion of finite groups H ≤ 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 elementary abelian p–groups if p odd and abelian 2–groups of exponent at most 4 if p = 2.