The gluing problem does not follow from homological properties of $\Delta_p(G)$

Abstract

Given a block $b$ in $kG$ where $k$ is an algebraically closed field of characteristic $p$, there are classes $\alpha_Q \in H^2 (Aut_\mathcal{F}(Q);k^\times)$, constructed by Külshammer and Puig, where $\mathcal{F}$ is the fusion system associated to $b$ and $Q$ is an $\mathcal{F}$-centric subgroup. The gluing problem in $\mathcal{F}$ has a solution if these classes are the restriction of a class $\alpha \in H^2(\mathcal{F}^c;k^\times)$. 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 $H^1_G(|\Delta_p(G)|;\mathcal{A}^1)$ vanishes, where $|\Delta_p(G)|$ is the simplicial complex of the non-trivial $p$-subgroups of $G$ and $\mathcal{A}^1$ is the coefficient functor $G/H \hookrightarrow \rm{Hom} (H, k^\times)$. The purpose of this note is to show that this group does not vanish if $G=\Sigma_{p^2}$ where $p \geq 5$.