Homotopy Representations of Classifying Spaces

Abstract

In the theory of compact Lie groups, the existence of faithful unitary representations for every compact Lie group is a consequence of the Peter–Weyl theorem. The existence of such representations imposes finiteness properties at the level of the cohomology of the classifying spaces. The proof involves analytic techniques which are not available for classifying spaces of homotopy theoretical structures such as p-compact groups and p-local finite groups. Understanding maps between classifying spaces is part of the program for developing an homotopy representation theory. In this paper I will describe progress made in this direction (joint work with L. Morales and J. Cantarero).