Homotopy properties of the poset of nontrivial p-subgroups of a group

Let G be a group and let Y,(G) be the set of nontrivial p-subgroups of G ordered by inclusion, where p is a prime. To the poset (= partially ordered set) YD(G) one can associate a simplicial complex 1 9JG)l in a well-known way. This simplicial complex appears in the work of Brown [2, 31 on Euler characteristics and cohomology for discrete groups. One of his results is an interesting variant of the Sylow theorem on the number of Sylow groups; it asserts that for G finite the Euler characteristic of 1 YD(G)I is congruent to 1 modulo the order of a Sylow p-subgroup. Our aim in this paper is to investigate various homotopy invariants of the simplicial complex ] YJG)l, such as its homology, connectivity, etc. We now review the contents of the paper. In Section 1 we review properties of the functor X ++ 1 X 1. Throughout the paper we use this functor to assign topological concepts to posets. For example, we call two posets homotopy equivalent when the associated simplicial complexes are. In Section 2 we first show that Y@(G) is homotopy equivalent to the poset z$(G) consisting of elementary abelian p-groups (called p-tori for short). Hence the remainder of the paper is mostly concerned with the smaller poset dJG). We show &‘,(Gr x G,) is h omotopy equivalent to the join of .z$(G1) and &JGa). We prove that dV(G) is contractible when G has a nontrivial normal p-subgroup, and state a conjecture to the effect that the converse holds when G is finite. The conjecture is proved for solvable groups in Section 12; the proof served as motivation for most of the second half of the paper. In Section 3 we show for a finite Chevalley group that -0lJG) has the homotopy type of the Tits building associated to G, and hence it has the homotopy type of a bouquet of spheres. Section 4 contains a proof of the aforementioned result of Brown. The next two sections relate the connected components of &(G) to topics in finite group theory, e.g., in Section 6, Puig’s analysis [S] of the Alperin fusion theorem is described.