The computation of the topological invariants of mapping spaces between Lie gorups

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Let X be a connected fiber of a Lie group (or finite H-space) G.By evaluating of the adjoint map G*OMEGA^nX*OMEGA^nX on cohomology, one can get information of the cohomology of the space of type Map (S^n, G). We investigate this problem by using a computer aided symbolic computation and the theoritical computation based Hopf algebra structures and cohomology operations. We got some results by the computer aided computation and could get more general results by the theoritical method.Especialy, for the case G is the Dwyer-Wilkerson's H-space, Spin (n) and Sp (n) (and the connected fibers of these spaces), we could evaluate the adjoint maps on the cohomology. These result show that the homotopy type of the space like Map (S^n, G) is ristricted strongly by these adjoint map.We got also the result about the commutator map and its lift to the connected fiber space. As an application of this, we can prove the unknown result about the homotopy commutativity of the rotation group.The symbolic caliculation like effective homology causes a hevy intermidiate explosion of computation and it makes very hevy garbage collection on the symbolic computaion system. About this problem, S.Matsui applied the method of parallel and conditional garbage collection and reduced GC time.K.Kawahigashi investigated the application to phisical phenomina which some properties of the low dimensional Lie group affects.