The concept of the C-space by F. Williams is generalized to the one defined on the category of higher homotopy associative //-spaces. This generalized concept is used to obtain the mod/? version of the torus theorem by J. Hubbuck.
Abstract. In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an An-space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected Ap-space has the finitely generated mod p cohomology for a prime p and the multiplication of it is homotopy commutative of the p-th order, then it has the mod p homotopy type of a finite product of Eilenberg-Mac Lane spaces K(Z, 1)s, K(Z, 2)s and K(Z/p i , 1)s for i ≥ 1.
Stasheff showed that if a map between H-spaces is an H-map, then the suspension of the map is extendable to a map between projective planes of the H-spaces. Stahseff also proved the converse under the assumption that the multiplication of the target space of the map is homotopy associative. We show by giving an example that the assumption of homotopy associativity of the multiplication of the target space is necessary to show the converse. We also show an analogous fact for maps between A nspaces.
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