The material in this Chapter is based on a lecture given by Peter Webb at the 1986 Arcata conference on Representation Theory of Finite Groups. I would like to thank him for supplying me with an early copy of the published version of this talk [282].

G-simplicial complexes

We shall be interested in group actions on topological spaces. Since we shall be interested in homotopical properties of group actions, and every topological space has the same weak homotopy type as its simplicial complex of singular chains, it is no real restriction to limit our attention to simplicial complexes with G acting simplicially, i.e., in such a way that the image of a simplex under a group element is always a simplex. Mostly our attention will be focused on the action of G on various finite simplicial complexes arising in a natural way from the subgroup structure of G. We shall be interested in representation theoretic invariants of these actions, and we shall concentrate on a definition for an arbitrary finite group of a generalised Steinberg module, which will be a virtual projective module which agrees for a Chevalley group (up to sign) with the usual Steinberg module.

Definition 6.1.1. Suppose G acts simplicially on a simplicial complex Δ. We say that Δ is aG-simplicial complexif whenever an element of G stabilises a simplex of Δ setwise, then it stabilises it pointwise.