Given a block b in kG where k is an algebraically closed field of characteristic p, there are classes αQ ∈ H2(AutF(Q);k×), constructed by Külshammer and Puig, where F is the fusion system associated to b and Q is an F-centric subgroup. The gluing problem in F has a solution if these classes are the restriction of a class α ∈ H2(Fc;k×). Linckelmann showed that a solution to the gluing problem gives rise to a reformulation of Alperin's weight conjecture. He then showed that the gluing problem has a solution if for every finite group G, the equivariant Bredon cohomology group H1G(|Δp(G)|;A1) vanishes, where |Δp(G)| is the simplicial complex of the non-trivial p-subgroups of G and A1 is the coefficient functor G/H ↦ Hom(H,k×). The purpose of this note is to show that this group does not vanish if G=Σp2 where p ≥ 5.
Homology, Homotopy and Applications, Vol. 12 (2010), No. 1, pp.1-10.
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