On group homomorphisms inducing mod-p cohomology isomorphisms

Equivariant CW-complexes and the orbit category

“…is an isomorphism if and only if H controls p-fusion in G (see [25], [33]). We have the following generalization (see Theorem 5.1) for functors of cohomological type over the orbit category (with respect to any family F ).…”

Mislin’s theorem for fusion systems through Mackey functors

“…A celebrated theorem of Mislin [8] shows that an isomorphism on mod-p cohomology (p a prime) implies control of p-fusion among compact Lie groups, and in particular among finite groups. Cartan and Eilenberg's stable elements theorem [5,XII.10.1] tells us that the mod-p cohomology ring H * (G, F p ) of a finite group G is isomorphic to the subring of the G-stable elements in the mod-p cohomology H * (S, F p ) of a Sylow p-subgroup S of G. In the language of fusion systems, this fact amounts to that H * (G, F p ) is determined by the fusion system F S (G) as a limit:…”

TheZ *-theorem for compact Lie groups

“…The main result of [Mi1] asserts that the converse also holds. (The Frobenius categories are called Quillen categories in [Mi1].)…”

“…The main result of [Mi1] asserts that the converse also holds. (The Frobenius categories are called Quillen categories in [Mi1].) We quote the full result for completeness, although we shall only use the easy part already mentioned for the proof of the Z * -theorem.…”

“…Our interest in the questions considered in this paper arose from a recent theorem of the first author [Mi1] giving a cohomological criterion for the control of finite p-fusion (see Theorem 1.1 below). However we do not need the full strength of this result here.…”