The gluing problem does not follow from homological properties of Δ p (G)

Given a block b in kG where k is an algebraically closed field of characteristic p, Given a block b in kG where k is an algebraically closed field of characteristic p , there are classes α Q ∈H 2 (Aut F (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∈H 2 (F c ;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 H 1 G (|Δ p (G)|;A 1 ) vanishes, where |Δ p (G)| is the simplicial complex of the non-trivial p -subgroups of G and A 1 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=Σ p 2 where p≥5 .