Alperin’s weight conjecture in terms of equivariant Bredon cohomology

摘要

Alperin’s weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in §1 and §2, respectively, and §3 provides a tool for computing the cohomology of the functors arising in §2. Taking as starting point the alternating sum formulation of Alperin’s weight conjecture by Knörr-Robinson [11], the material of the previous sections is applied in §4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [15, 16, 17].