Webb's conjecture for fusion systems.
type: Item
instance of: scholarly article
title: Webb's conjecture for fusion systems.
zbMATH Open document ID: 1177.20017
author: Assaf Libman
published in: Israel Journal of Mathematics
publication date: 2009-02-17T00:00:00Z
review text: The partially ordered set of all \(p\)-subgroups of a finite group \(G\) induces a simplicial complex coming with a \(G\)-action. The orbit space is \(\mathbb{F}_p\)-acyclic, as was shown by Webb, and Webb's conjecture states that it is actually contractible. The conjecture was shown by Symonds in a slightly more general form. The purpose of the paper is to formulate the conjecture for fusion systems, and to prove the conjecture for fusion systems. The proof is very elegant, using mainly one observation, linking weak homotopy equivalence on fixed point spaces to weak homotopy equivalences of orbit spaces. The paper generalises and simplifies in particular a result of Linckelmann, who proved Webb's conjecture for fusion systems around the same time as well, along the lines of Symonds' original proof.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 55U15
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5508704
zbMATH Keywords: Webb conjecture
zbMATH Keywords: fusion systems
zbMATH Keywords: block cohomology
zbMATH Keywords: weak homotopy equivalences
MaRDI profile type: Publication
full work available at URL: s11856-008-1044-8
OpenAlex ID: W2088697896
Wikidata QID: Q122898961
cites work: Homotopy limits, completions and localizations
cites work: The homotopy theory of fusion systems
cites work: Euler characteristics of discrete groups and G-spaces
cites work: High dimensional cohomology of discrete groups
cites work: Simplicial homotopy theory
cites work: The orbit space of a fusion system is contractible
cites work: Homotopy properties of the poset of nontrivial p-subgroups of a group
cites work: Amalgams, blocks, weights, fusion systems and finite simple groups.
cites work: The orbit space of the \(p\)-subgroup complex is contractible
cites work: Homotopy equivalence of posets with a group action
cites work: Transformation groups
DOI: 10.1007/S11856-008-1044-8
description: scientific article; zbMATH DE number 5508704