Classifying spaces of Categories, Flore Homology and Homotopsy Theory of Infinite Complexes

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The most important results of the research project are the following :(1) Topology of infinite dimensional Lie groups(2) Morse theory of infinite dimensional manifolds(3) Classifying space of categories and dynamical systemsFor (1) we determined homology ring of free loop groups for 1-connected Lie groups completely. Using the result we are nou determining the cohomology of the classifying spaces of them. For gauge groups, we showed the case when the base spaces are closed 1-connectedmanifolds and the structure group is SU(2), the homotopy type of the classifying spaces completely determine the boinotopy type of the base space and the isomorphism class of the bundles.For (2) K. Fukaya solved the Anord conjecture using the Gromov-Witten iavariants.For (3) we considered the Conley homology. We considerd a certain category and the cohomology of the classifying space of it is isomorphic to the Conley homology under some fine coditions.