On the homotopy type of the Quillen complex of finite soluble groups (Q1763744)

Given a finite group \(G\) and a prime \(p\), let \({\mathcal A}_p(G)\) denote the poset (partially ordered set) of all non-trivial elementary abelian \(p\)-subgroups of \(G\) and \(\Delta ({\mathcal A}_p(G))\) the Quillen complex of \(G\) given by the ordered complex of the poset \({\mathcal A}_p(G)\). In this paper the author studies the homotopy type of \(\Delta ({\mathcal A}_p(G))\). In particular, he investigates some techniques of the theory of diagrams and homotopy colimits (cf. [\textit{J. Pulkus} and \textit{V. Welker}, J. Austral. Math. Soc. Ser. A 69, 212--228 (2000; Zbl 0986.20022)] and \textit{G. M. Ziegler} and \textit{R.T. Zivaljeviź}, Math. Ann. 295, 527--548 (1993; Zbl 0792.55002)]) and he proves homotopy equivalence to a wedge of spheres of possible different dimensions when \(G\) is a finite soluble group and \(p\not= 2\).