Homology, Homotopy and Applications

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

Assaf Libman

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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 $p \geq 5$.

Article information

Source
Homology Homotopy Appl., Volume 12, Number 1 (2010), 1-10.

Dates
First available in Project Euclid: 28 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.hha/1296223817

Mathematical Reviews number (MathSciNet)
MR2594678

Zentralblatt MATH identifier
1196.20014

Subjects
Primary: 20C20: Modular representations and characters 55N25: Homology with local coefficients, equivariant cohomology 05E25

Keywords
Gluing problem Alperin’s conjecture equivariant cohomology

Citation

Libman, Assaf. The gluing problem does not follow from homological properties of $\Delta_p(G)$. Homology Homotopy Appl. 12 (2010), no. 1, 1--10. https://projecteuclid.org/euclid.hha/1296223817


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