Some homotopy equivalences for sporadic geometries

Stephen D. Smith and Satoshi Yoshiara

To appear in J. Algebra

Abstract

A previous work [RSY90] established 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 let to a more unified treatment---
establishing the stronger result of homotopy equivalence of the $p$-local
geometry with the usual poset $\mathcal A_p(G)$. 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 co-incides with those of characteristic $p$-type: suggesting 
another equivalence method via the poset $\mathcal B_p(G)$ of radical (or
stubborn) $p$-subgroups. In particular, one finds that these sporadic
groups satisfy an analog of the Borel-Tits theorem---that normalizers of
$p$-groups lie in vertex stabilizers. Still further intriguing coincidences
remain to be explained.