Department of Mathematics
"Homology decompositions for classifying spaces of finite groups, Part III"
Abstract: In this third seminar of the series, I will prove the ampleness and sharpness of certain collections of subgroups in a finite group G with respect to some homology decompositions. This will complete our discussion on homology decompositions for finite groups using the approach due to Dwyer. For a prime p, dividing the order of G, a mod-p homology decomposition of the classifying space BG is a mod-p homology isomorphism between BG and a homotopy colimit of a functor F from a category D to the category of spaces, such that for each object d in D, F(d) is homotopy equivalent to BH_d for some subgroup H_d of G. There are mainly three types of homology decompositions: subgroup decomposition, centraliser decomposition, and normaliser decomposition. In all these decompositions, the category D is described in terms of an ample collection C of subgroups. A collection C is ample if mod-p equivariant cohomology of the topological realization of the poset of subgroups in C, is isomorphic to the mod-p cohomology of the group G. The collection of all nontrivial p-subgroups, the collection of all nontrivial elementary abelian p-subgroups, and the collection of all p-centric and p-radical subgroups are ample collections. Some of these ample collections are also sharp with respect to some of the homology decompositions (but not all). These results are due to Webb, Jackowski, McClure, Oliver, Dwyer, and others.
Date: Monday, April 18, 2016
Place: Mathematics Seminar Room, SA-141
All are most cordially invited. Tea and cookies will be served after the talk.