Study of the higher homotopy commutative Hopf spaees and its application to the higher category

在此期间，我们得到了以下结果。研究了Williams给出的c_n结构的各种类型的同伦交换性，并将它们的多面体看作是复面体的子集。特别地，我们证明了由若干固定洗牌的逆组成的第n个对称群的子集所给出的表示高同伦交换性的多映体是洗牌多映体。在a_n空间上引入了ac_n结构的概念，它是一种高同伦交换性。此外，我们还研究了n>(p-1)/2下，简化幂运算对具有ac_n结构的模p上同有限空间的作用。该研究结果发表在《几何与拓扑专论》上。我们确定了奇素数p在单连通无扭转的有限p紧群上的约简幂运算的作用，并确定了空间作为原子空间积的分解。结果推广了mimura - nishida - toda关于单连通单李群的事实。引入了拓扑一元的高同伦交换性的新概念。为了做到这一点，我们使用了由Gelfand， Kapranov和Zelevinsky构建的结果图。我们的研究与Felix-Tanre和Aguade所提出的概念有着深刻的关系。

We have the following result during the period.1. We study various types of homotopy commutativity related to the C_n-structure given by Williams, and we consider the polytopes representing them as subsets of the permutohedron. In particular, we show that the polytopes representing the higher homotopy commutativity given by the subset of n-th symmetric group consists of the inverse of some fixed shuffles are shuffle polytopes.2. We introduce the concept of AC_n-structure on A_n-spaces, which is a higher homotopy commutativity. Moreover, we study that the action of the reduced power operation on the mod p cohomology finite A_p-space with AC_n-structure for n>(p-1)/2. The result is published in Geometry and Topology Monographs.3. We determine the action of the reduced power operation on the simply connected finite p-compact groups with no torsion in homology for odd primes p. We also determine the decomposition of the spaces as products of atomic spaces. The result is a generalization of the fact on the simply connected simple Lie groups given by Mimura-Nishida-Toda.4. We introduced a new concept on higher homotopy commutativity of topological monoids. To do so we used resulthedra which are constructed by Gelfand, Kapranov and Zelevinsky. Our study has a deep relation ship with the concepts given by Felix-Tanre and Aguade.