Some Homotopy Equivalences for Sporadic Geometries

Abstract

A previous work [RSY90] established the projectivity of the reduced Lefschetz modules of certain sporadic group geometries, and the present paper continues that work in a wider context.Recent developments in sporadic-group cohomology include some applications of [RSY90], which in turn suggested treatment of a broader class of geometries. Recurring similarities in the proofs also led to a more unified treatment—establishing the stronger result of homotopy equivalence of thep-local geometry with the usual elementary poset Ap(G). One equivalence method proceeds by means of a new “closed set” in a standard technique of Quillen. It was further observed that the larger list of simple groups now treated essentially coincides with those of characteristicp-type, suggesting another equivalence method via the poset Bp(G) of radical (or stubborn)p-subgroups. In particular, one finds that these sporadic groups satisfy an analogue of the Borel–Tits theorem—that normalizers ofp-groups lie in simplex stabilizers. Still further intriguing coincidences remain to be explained.