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

设X是李群（或有限h空间）G上的连通光纤，通过求上同调上的伴随映射G* * ^nX* * ^nX，可以得到map （S^n， G）型空间的上同调信息。我们利用计算机辅助符号计算和基于Hopf代数结构和上同调运算的理论计算来研究这一问题。通过计算机辅助计算得到了一些结果，通过理论方法可以得到更一般的结果。特别地，对于G是Dwyer-Wilkerson的h空间，Spin (n)和Sp (n)（以及这些空间的连通纤维），我们可以在上同调上计算伴随映射。这些结果表明类空间映射（S^n， G）的同伦型受到这些伴随映射的强烈限制。我们还得到了换向子映射及其到连接光纤空间的提升的结果。作为应用，我们证明了旋转群的同伦交换性的未知结果。像有效同调这样的符号计算导致了大量的中间计算量的爆炸，并且对符号计算系统造成了非常严重的垃圾回收。针对这一问题，S.Matsui采用了并行和条件垃圾回收的方法，减少了GC时间，k.k awahigashi研究了低维李群的某些性质对物理现象的影响。

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.