The gluing problem does not follow from homological properties of ${\Delta}_p(G)$

Given a block $b$ in $kG$ where $k$ is an algebraically closed field of characteristic $p$, there are classes $\alpha_Q \in H^2(\mathrm{Aut}_{\mathcal{F}}(Q);k^\times)$, 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 $G/H \mapsto \mathrm{Hom}(H,k^\times)$. The purpose of this note is to show that this group does not vanish if $G=\Sigma_{p^2}$ where $p\geq 5$.

Keywords

gluing problem, Alperin’s conjecture, equivariant cohomology

2010 Mathematics Subject Classification

05Exx, 20C20, 55N25

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Published 1 January 2010