turkmath.org

This will be a survey talk on the close relationship between the local structure of a finite group or compact Lie group and that of its classifying space. By the "p-local structure" of a group G, for a prime p, is meant the structure of a Sylow p-subgroup S of G (a maximal p-toral subgroup if G is compact Lie), together with all G-conjugacy relations between elements and subgroups of S. By the p-local structure of the classifying space BG is meant the structure (homotopy properties) of the p-completion of BG. For example, by a conjecture of Martino and Priddy, now a theorem, two finite groups G and H have equivalent p-local structures if and only if the p-completions of BG and BH are homotopy equivalent. This (the "if" part of the statement) was used, in joint work with Broto and M{\o}ller, to prove a general theorem about local equivalences between finite Lie groups --- a result for which no purely algebraic proof is known. As another example, these ideas have allowed us to extend the family of p-completed classifying spaces of (finite or compact Lie) groups to a much larger family of spaces which have many of the same very nice homotopy theoretic properties.