Homotopy Theoretic Methods in Group Cohomology
The volume contains notes on two series of lectures delivered during the Advanced Course on Classifying Spaces and Cohomology of Groups at the Centre de Recerca Mathemàtica in Bellaterra. The first series of lectures was written by W.G. Dwyer (Classifying Spaces and Homology Decomposition), and the second one by H.-W. Henn (Cohomology of Groups and Unstable Modules over the Steenrod Algebra). The first series deals with a homology decomposition of the classifying space of a finite group as a homotopy colimit of classifying spaces of some of its subgroups. By homology decomposition we mean a mod p homology isomorphism between the colimit and the classifying space. The second series discusses a broader variety of groups (e.g., compact Lie groups, arithmetic groups, mapping class groups, etc). It is based on non-stable modules over the Steenrod algebra, in particular, on the Lannes functor TV. It is exactly the calculation of the degree 0 component of TVH*BG, which coincides with the Quillen theory of F-isomorphisms. The main results show that the knowledge of TVH*BG (even a partial one) enables us to discover various approximations to H*BG. The text is designed first of all for postgraduate students. It is relatively short and students will have to look at original sources for some details. Nevertheless, I think that for a beginner in the area, the book is an excellent introduction.