Simple connectivity of the Quillen complex of the symmetric group. (Q1406740)

Given a finite group \(G\) and a prime \(p\), the Quillen complex of \(G\), denoted \(\Delta A_p(G)\) is the order complex of the poset \(A_p(G)\) of nontrivial elementary Abelian \(p\)-subgroups of \(G\), with \(G\) acting by conjugation on \(\Delta A_p(G)\). As noted, the simplicial complex \(\Delta A_p(G)\) is mostly connected if its dimension is \(\geq 1\), with several known low-dimensional exceptions. Replacing connected by simply connected leads to a new set of perspectives, a new set of questions and a new set of some answers, including the main one of this interesting paper that \(\Delta A_p(S_n)\) (\(=\Delta A_p(A_n)\) \(p\) is odd, \(A_n\) simple for \(n\geq 5\)) is simply connected if \(p\) is odd and \(3p+2\leq n\leq p^2\) or \(n\geq p^2+ p\), through a very detailed and careful study of connections between \(A_p(S_n)\) and \(T_p(n)\) the subposet of \(A_p(S_n)\) formed by elementary Abelian \(p\)-subgroups generated by \(p\)-cycles, as well as interesting new information generated on both these types of posets whence also on their various complexes including the Quillen complex.