The Homotopy Theory of Classifying Spaces

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DMS-0104318Jesper GrodalThe project consists of developing the homotopy theory of classifying spaces of compact Lie groups (including finite groups) and their generalizations. The goal of this is to solve central problems in homotopy theory involving classifying spaces, as well as to discover new relationsto problems in group theory and representation theory. A further goal is to produce results in related fields such as in the theory of group actions, equivariant cohomology, and group cohomology. The project centersaround developing the homotopy theoretic group theory of the classifying space BG, or rather of its p-completion. This involves examining the induction theory, p-local theory, representation theory as well as the more general actions of these objects. The methods used come from modern unstable homotopy theory, developed since the solution of the Sullivan conjecture, combined with ideas and techniques from modern group theory.Forming the classifying space BG of a group of symmetries G is a way of geometrically or topologically encoding the "symmetries" present in G. Geometric spaces of this form and generalizations thereof are ubiquitous in topology, and hence form an important class of spaces to study. Furthermore, the less rigid structure of the classifying space BG, compared to that ofthe group G, makes certain techniques available in the study of the former which were not available when dealing with the later directly. Hence, studying the grouptheory of BG provides an interesting and useful way of studying the original group G.