Hypergraph matching complexes and Quillen complexes of symmetric groups. (Q1826947)

Consider the symmetric group \(S_n\). Given a prime \(p\), the Quillen complex of \(S_n\) at \(p\) is a simplicial complex formed by taking the order complex of the poset of non-trivial elementary Abelian \(p\)-subgroups of \(S_n\). The goal of this paper is to study the homotopy and homology of such Quillen complexes building on work done by \textit{R. Ksontini} [Propriétés homotopiques du complexe de Quillen du group symétrique, Thèse de doctorat, Université de Lausanne (2000)]. The basic strategy is to relate the topology of the Quillen complex to that of two other simplicial complexes: the \(p\)-cycle complex and the hypergraph matching complex. The face poset of the \(p\)-cycle complex consists of those elementary Abelian \(p\)-subgroups generated by \(p\)-cycles. And the face poset of the hypergraph matching complex consists of the poset of partitions of \([n]\) into subsets of size \(1\) and \(p\). Ksontini showed that when \(n<p^2\), the Quillen complex is homotopy equivalent to the \(p\)-cycle complex. For \(p^2\leq n<p^2+p\), one must add a wedge of one-spheres. In this work, the author extends these results. The first case considered is when \(n=3p\) for \(p>3\). It is shown that the Quillen complex is again homotopy equivalent to the \(p\)-cycle complex, and the homotopy type is precisely identified as a wedge of two-spheres and one-spheres. Next, the author considers the case when \(p^2 +p\leq n<2p^2\) for \(p\geq 3\). Here the results are less precise. The Quillen complex is homotopy equivalent to the \(p\)-cycle complex wedged with the join of the \(p\)-cycle complex for \(S_{n-p^2}\) (rather than \(S_n\)) with a wedge of one-spheres. When \(n-p^2\) is sufficiently small, this additional factor can be more precisely identified. A general consequence of this homotopy equivalence is the fact that the homology of the \(p\)-cycle complex is a summand of the homology of the Quillen complex. In certain cases the homology of the \(p\)-cycle complex can be identified as a sum of homologies of hypergraph matching complexes. Thanks to a computer computation of J.-G. Dumas of homology for hypergraph matching complexes, this leads to the quite interesting existence of a torsion class in the second cohomology of the Quillen complex for \(S_{13}\) at \(p=3\). As noted by the author, this seems to be the first known example of a torsion element and it would be nice to have a formal proof of the existence of this class.