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

在此期间，我们取得了以下结果1.我们研究了Williams给出的与C_n-结构相关的各种类型的同伦交换性，并将表示它们的多面体看作置换面体的子集。特别地，我们证明了表示由第n个对称群的子集所给出的高同伦交换性的多面体是由一些固定洗牌的逆构成的。在A_n-空间上引入了AC_n-结构的概念，它是一种较高的同伦交换性。此外，我们还研究了约化幂运算在n&gt；(p-1)/2的具有AC_n结构的mod_p上同调有限A_p-空间上的作用。结果发表在《几何与拓扑专著》上。确定了奇素数p的同调无挠的单连通有限p-紧群上的约化幂运算的作用，并确定了空间的分解为原子空间的乘积.其结果推广了Mimura-Nishida-Toda关于单连通单李群的一个事实。引入了拓扑么半群的高次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Zlevinsky构造的结果dra。我们的研究与Felix-Tanre和Aguade提出的概念有很深的关系。

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.