Some results about Quillen complex of \(\text{Sp}_{2n}(q)\) (Q1277009)

Let \(G\) be a finite group and let prime \(p\) divide \(|G|\). The Quillen complex of \(G\) at \(p\) is the order complex of the poset of nontrivial elementary Abelian \(p\)-subgroups of \(G\) (ordered by inclusion) and is denoted by \({\mathcal A}_p(G)\). It is well-known that \({\mathcal A}_p(G)\) is disconnected if and only if \(G\) contains a strongly \(p\)-embedded subgroup. A result of Gorenstein and Lyons classifies the pairs \((G,p)\) for which \({\mathcal A}_p(G)\) is disconnected. Thus it is natural to attempt to characterize the pairs \((G,p)\) for which \({\mathcal A}_p(G)\) is simply connected. A result of Aschbacher has reduced this question to a minimal class of groups. The main result of this paper contributes to this problem: Theorem A. Let \(q\) be a prime power and let \(p\) be a prime such that \(p\mid(q-1)\). Then \({\mathcal A}_p(\text{Sp}_{2n}(q))\) is Cohen-Macaulay of dimension \(n-1\). If \(d\) is the order of \(q\) in \(\mathbb{Z}/p\mathbb{Z}\) and \(d\geq 3\) and \(d\) is odd, then \({\mathcal A}_p(\text{Sp}_{2n}(q))\) is simply connected when \(m_p(\text{Sp}_{2n}(q))\geq 3\). This result has the following Corollary: Theorem B. Let \(V\) be a \(2N\)-dimensional symplectic space over \(\mathbb{F}_q\) with \(q\neq 2\) and let \(\mathcal C\) be the order complex of the proper nondegenerate subspaces of \(V\) (ordered by inclusion). Then \(\mathcal C\) is Cohen-Macaulay of dimension \(n-2\).