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

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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.