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Barcelona Algebraic Topology Group

• • • • • • • • • • Aplicacions entre espais classificadors de grups de Kac-Moody de rang 2

University: Status: Universitat Autònoma de Barcelona Finished Phd 02-07-2001 Digital version The study of the maps between classifying spaces of compact Lie groups has been one of the highlights of algebraic topology in the final quarter of the XXth century.From a simply connected compact connected Lie group we obtain a finite dimensional Lie algebra, and from a finite dimensional Lie algebra we obtain a Cartan matrix. A Cartan matrix A=(ai,j) is a positive definite matrix with integer coefficients such that ai,i=2, ai,j ≥ 0 and ai,j=0 implies aj,i=0.All this proces can be inverted and we can recover the Lie algebra from the Cartan matrix and ``integrate'' this Lie algebra to obtain a simply connected, compact, connected Lie group.Consider now a generalizad Cartan matrix, that is a non necessarily positive definite Cartan matrix. We can construct an integrable Lie algebra (non finite dimensional, in general) and from it a topologicla group. The result of these constructions are the called Kac-Moody algebras and Kac-Moody groups.From a homotopy point of view the Kac-Moody groups were studied by N. Kitchloo (cohomological properties) and thisresults took us to consider other well known results in compact Lie groups to be generalized to Kac-Moody groups.The main result of the thesis is the study of the mapping space [BK,BK], where K is a rank 2 Kac-Moody group.In order to understand [BK,BK] we must calculate [BT,BK], where T is a maximal torus in K. Here we get results which do not agree with the compact Lie group case: there exist maps from BT to BK which do not come from representations.With this study we get a complete description of the subspace of [BT,BK] which involve all the maps which come from maps of [BK,BK]. This classification will allow us to understand the space [BK,BK],  after proving that the map induced by the inclusion [BK,BK] in [BT,BK] is injective.Studying this we get other results like the characterization of the homotopy type of the rank 2 Kac-Moody groups (in particular we obtain non-isomorphic Kac-Moody groups with the same classifying space) and a characterization of the possible degrees of maps from BK to BK.