Department of Mathematics
"Homology decompositions for classifying spaces of finite groups, Part I"
Abstract: In this series of talks, I will explain the theory of mod-p Homology decompositions for finite groups using an 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 realisation 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. We will also discuss sharpness of these collections with respect to different types of homology decompositions.
These results are due to Webb, Jackowski, McClure, Oliver, Dwyer, and others. In the last talk of the series, I will discuss some applications of homology decompositions to constructing group actions on homotopy spheres.
Date: Monday, April 4, 2016
Place: Mathematics Seminar Room, SA-141
All are most cordially invited. Tea and cookies will be served after the talk.