Teoria homotòpica de Grups de Lie

We study Lie groups, particularly, finite groups, compact connected Lie groups and Kac-Moody groups, from the point of view of homotopy theory. For this aim, one translates algebraic or geometric properties of groups into homotopy theoretic properties of the classifying spaces. Next, one isolates one prime at a time by means of Bousfield-Kan p -completion. p -compact groups are homotopic theoretic generalizations of compact Lie groups. Its definition adn quick development of ideas followed Miller's solution of the Sullivan conjecture and Lanne's theory. We are interested in the normalizer conjecture of M\philler, the investigation of particular classes of spaces of maps between classifying spaces of p -compact groups, and of certain classes of embedings or representations of (exotic) p -compact groups. We plan to extend the methods of p -compact groups in order to study homotopy uniqueness of p -completed classifying spaces of finite groups and the homotopy type of topological monoids of self homotopy equivalences of p -completed classifying spaces of finite groups. We also investigate Kac-Moody groups from this same point of view. Kac-Moody groups, usually infinite dimensional, are combinatorial generalizations of Lie groups and behave like Lie groups in many respects. Though, very little is known about their topological properties. We are interested in the homotopy type of their classifying spaces