The orbit space of a fusion system is contractible

Abstract

Given a fusion system ℱ on a finite p-group P, where p is a prime, we show that the partially ordered set of isomorphism classes in ℱ of chains of non-trivial subgroups of P, considered as topological space, is contractible, further generalising Symonds’, proof [Comment. Math. Helvet. 73 (1998) 400–405] of a conjecture of Webb [Comment. Math. Helvet. 66 (1991) 34–69; Arcata Conference on Representations of Finite Groups, part I, Proceedings of Symposia in Pure Mathematics 47 349–365] and its generalisation to non-trivial Brauer pairs associated with a p-block by Barker [J. Algebra 212 (1999) 460–465].