Topologs of manifolds and mathematical plysics

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The reserch results supported by Grants in Aid for Scientific Reserch (C) 05640105 consist of the following1. Free loop group : det G be a compact, connected diegroup, LAMBDAG the space of free loops on G and p a prine. Then the following two conditions are equrialent :(1) The integral cohomology of G is p-toscn free(2) H^* (BLAMBDAG : */p)*H^* (BG : */p) <cross product> H^* (G : */p) as an algeha.a similon resutt for the Dwyer-Wilkerson H-space is also obtained.2. Infinite dimensional die groups : Topolegs of infinite dimesional die group (gauze groups)3. Morse homotopy type : det M be a conysact manifold and f : M*IR a Morse function. The homotopy type of M can be cletermined by f.Fukaya dofined a categary by f and described the quantum homotopy type of a syinplectic monifolds. I oftainea a method to describe the homotopy type of M by f, Morsover Ueno studiea confomal field theoy using algehaie geometory and Maruyama studied classifying spaces, moduli of verion connections and modubi of vectr bmdles.