On Quillen's conjecture for \(p\)-solvable groups (Q1786464)

Let \(G\) be a finite group, let \(p\) be a prime and let \(\mathcal A_p(G)\) be the poset consisting of the non-trivial elementary abelian \(p\)-subgroups of \(G\) ordered by inclusion. The homotopy properties of the topological realization \(|\mathcal A_p(G)|\) were first studied by \textit{D. Quillen} [Adv. Math. 28, 101--128 (1978; Zbl 0388.55007)], who in particular introduced the following conjecture: if \(|\mathcal A_p(G)|\) is contractible then \(O_p(G)=1.\) In order to prove Quillen's conjecture, it suffices to prove that it \(G\) has \(p\)-rank \(r\) and \(O_p(G)=1,\) then \(\tilde H_{r-1}(|\mathcal A_p(G)|; \mathbb Q) \neq 0\). This was proven by Quillen [loc. cit.] when \(G\) is solvable and by \textit{J. L. Alperin} [Lect. Notes Math. 1456, 1--9 (1990; Zbl 0718.20008)], via the Classification of the Finite Simple Groups (CFSG), for \(G\) \(p\)-solvable. In this paper, the author provides new proofs for the solvable and the \(p\)-solvable cases of Quillen's conjecture. He also gives an asymptotic version for the \(p\)-solvable case that does not use the CFSG. The arguments are of geometric nature. The non-zero top dimensional homology class constructed by the author belongs to a suitable subgroup of \(\tilde H_{r-1}(|\mathcal A_p(G)|; \mathbb Q)\) whose elements involve only the top two layers of the poset \(\mathcal A_p(G),\) i.e. Sylow subgroups and their hyperplanes.