Suppose X is a simply connected mod p H-space such that the mod p cohomology H ((X) is a nitely generated algebra. We show that the loop space X is homotopy equivalent to a nite product of Eilenberg-MacLane spaces K ; 1) for i 1. This is a generalization of the result due to Lin, in which the same result was proved under the assumption that X is an A p-space.

We show that Szczarba's twisting cochain for a twisted Cartesian product is essentially the same as one constructed by Shih. More precisely, can be obtained via basic perturbation lemma if uses 'reversed' version of classical Eilenberg-MacLane homotopy Eilenberg-Zilber contraction. Along way we prove several new identities involving these homotopies.

Let X be a connected space and let K = H∗(X; Fp) where p is an odd prime. We construct functors ω and l which approximate the cohomology of the free loop space ΛX as follows: There are morphisms ω(K) → H∗(ΛX; Fp) and l(K) → H ∗ S1(ΛX;Fp). These are isomorphisms when X is a product of Eilenberg-MacLane spaces of type K(Fp, n) for n ≥ 1. MSC: 55N91; 55P35; 55R12

We give a particular choice of the higher Eilenberg-MacLane maps by a recursive formula. This choice leads to a simple description of the homotopy operations for simplicial Z/2-algebras.

The isomorphism of 0-homology groups of a categorical at zero semigroup and homology groups of its 0-reflector is proved. Some applications of 0-homology to Eilenberg–MacLane homology of semigroups are given.

We compute the topological complexity of Eilenberg-MacLane spaces associated to the group of automorphisms of a finitely generated free group which act by conjugation on a given basis, and to certain subgroups.