Simple connectivity of the Quillen complex of the symmetric group.
type: Item
instance of: scholarly article
title: Simple connectivity of the Quillen complex of the symmetric group.
zbMATH Open document ID: 1040.20018
DOI: 10.1016/S0097-3165(03)00073-6
author: Rached Ksontini
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-09-07T00:00:00Z
review text: 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.
reviewed by: J. Neggers
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55N99
zbMATH DE Number: 1975834
zbMATH Keywords: subgroup complexes
zbMATH Keywords: Quillen complex
zbMATH Keywords: symmetric group
zbMATH Keywords: simple connectivity
zbMATH Keywords: elementary Abelian subgroups
zbMATH Keywords: homotopy type
MaRDI profile type: Publication
cites work: Simple connectivity of \(p\)-group complexes
cites work: Homology of certain sets of 2-subgroups of symmetric groups
cites work: The fundamental group of the Quillen complex of the symmetric group
cites work: Homotopy properties of the poset of nontrivial p-subgroups of a group
cites work: Q3994730
cites work: Q3794385
description: scientific article; zbMATH DE number 1975834