Higher Homotopy Commutativity of $H$-spaces and the Cyclohedra

Abstract

We define a higher homotopy commutativity of $H$-spaces using the cyclohedra $\{W_n{\}}_{n\ge 1}$ constructed by Bott and Taubes. An $H$-space whose multiplication is homotopy commutative of the $n$-th order is called a $B_n$-space. We also give combinatorial decompositions of the permuto-associahedra $\{K{P}_n{\}}_{n\ge 1}$ introduced by Kapranov into union of product spaces of the cyclohedra. From the decomposition, we have a relation between the $B_n$-structures and another higher homotopy commutativity represented by the permuto-associahedra.