Alperin's weight conjecture in terms of equivariant Bredon cohomology

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 Knorr-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].