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

在此期间，我们有以下结果。本文研究了与威廉姆斯给出的C_n-结构有关的各种同伦交换性，并把表示它们的多面体看作置换面体的子集。特别地，我们证明了由某些固定shuffle的逆构成的n阶对称群的子集所给出的表示高阶同伦交换性的多面体是shuffle多面体.本文在A_n-空间上引入AC_n-结构的概念，它是一种高阶同伦交换性。进一步研究了约化幂运算在具有AC_n-结构的模p上同调有限A_p-空间（n>（p-1）/2）上的作用。结果发表在《几何与拓扑学专著》上。我们确定的行动，约化功率操作的单连通有限p-紧群，没有扭转的同调奇素数p。我们还确定了分解的空间作为产品的原子空间。这个结果是Mimura-Nishida-Toda关于单连通单李群的结果的推广.引入了拓扑幺半群的高同伦交换性的新概念。要做到这一点，我们使用的resultethedra是由Gelfand，Kapranov和Zelevinsky。我们的研究与Felix-Tanre和Aguade提出的概念有很深的联系。

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.