Webb's conjecture for fusion systems. (Q1001404)

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.