Teoria homotòpica de grups finits, grups de Lie i grups de Kac-Moody

We study Lie groups, particularly finite groups, compact and connected Lie groups and Kac-Moody groups from the point of view of homotopy theory. For this aim, algebraic or geometric properties of the group are translated into homotopy theoretic properties of its classifying sapce. Then, one isolates one prime at a time by means of the Bousfield-Kan p-completion. We study homotopy representationts and homology decompositions of p-compact groups. These are generalizations in homotopy theory of compact connected Lie groups. The construction of a suitable homotopy theory of finite groups is planed. It will require the definition of new objects, called homotopy finite groups, that will recover axiomatically the esentials of the p-local structure of finite groups. We will investigate exotic cases like Sol(q) and other homotopic models that generalize the concept of Chevalley group, and that are expected to fall into this new class of objects. We plan to study generalizations of the theory of p-compact groups to a theory of homotopy Lie groups, where there will be no extra restrictions on the groups of components. To this aim we take the theories of p-compact groups and of homotopy finite groups as the starting point. A theory of extensions will be needed. We also investigate infinite discrete groups and Kac-Moody groups from the same optic. Kac-Moody groups, usually infinite dimensional groups, are combinatorial generalizations of compact and connected Lie groups. Kac-Moody groups behave like Lie groups in many respects. However, little is known about its topological properties. We are interested in the homotopy type of the classifying spaces, and maps between them, including Adams operations