Homology decompositions from subgroups complexes of finite groups

  Stephen D. Smith
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago

In the mid 1970s Brown and especially Quillen studied homotopy properties of simplicial complexes determined by collections of p-subgroups of a finite group G: a particularly striking example is the case of G a group of Lie type in characteristic p, when a natural such complex is equivalent to the geometry given by the Tits building for G. Webb further developed the properties, with emphasis on sufficient conditions on complexes to guarantee an alternating-sum decomposition for the group cohomology of G; intriguing examples were given by the ``p-local geometries'' (somewhat analogous to buildings) being studied by finite group theorists during the early 1980s, for such examples as sporadic simple groups G. In the intervening years there has been increasing interaction in this area between algebraic topology and finite group theory. The talk will survey some of the development through the present (and near future?).