Stephen D. Smith: Poset combinatorics and cohomology of sporadic simple groups

The work of Tits on buildings, and of Brown and Quillen on posets of p-subgroups of general finite groups, led after about 1978 to a vigorous development of research on complexes of subgroups -- crossing the boundaries of algebraic topology, combinatorics, and finite group theory.

Work of Webb (following Brown and Quillen) gave conditions on a poset of subgroups of a general finite group, in order to guarantee a decomposition of group cohomology -- in terms of the cohomology of various poset elements. "Decomposition theory" was further developed in the context of homotopy theory, by Jackowski, McClure, Oliver, Dwyer, Grodal, and others.

The methods were applied in particular to SIMPLE groups, by various authors; and from the early 1990s, Benson and Smith began a project to exhibit such a decomposition (at p=2) for each of the 26 sporadic simple groups -- over a particularly natural finite geometry afforded by 2-local subgroups. The project was completed in 2008, with the appearance of Vol 147 of AMS Surveys.

The talk will give an exposition of the area and the result, with an emphasis on the essentially combinatorial content of the methods.