Control of fusion and cohomology of trivial source modules

Abstract

Let G be a finite group and H a subgroup. We give an algebraic proof of Mislin's theorem which states that the restriction map from G to H on mod-p cohomology is an isomorphism if and only if H controls p-fusion in G. We follow the approach of P. Symonds [P. Symonds, Mackey functors and control of fusion, Bull. London Math. Soc. 36 (2004) 623-632] and consider the cohomology of trivial source modules.

A theorem of Mislin for cohomology of fusion systems and applications to block algebras of finite groups

“…A celebrated theorem of Mislin in [15] regarding the control of fusion in group cohomology (stated for compact Lie groups) has now a new short algebraic proof for p odd thanks to Benson, Grodal and Henke [4]. See [7], [19] for other algebraic proofs which uses Mackey functors and cohomology of trivial source modules; see also [21] for a different algebraic approach. Also, in [8,Remark 5.8] Linckelmann suggests a topological proof for Mislin's theorem in the case of block algebras of finite groups, more precisely for cohomology of fusion systems associated to blocks.…”

Group cohomology and control of p-fusion

“…Symonds [41], following an idea of Robinson [39, §7], provided an algebraic reduction of the problem to a statement about cohomology of trivial source modules, which he then proved topologically. Algebraic proofs were finally completed independently by Hida [21] and Okuyama [33], who gave algebraic proofs of Symonds' statement, through quite delicate arguments in modular representation theory. (See also e.g., [1] and [42].…”