### Categories equivalent to the category of rational H-spaces.

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We give conditions for a map of spaces to induce maps of the homology decompositions of the spaces which are compatible with the homology sections and dual Postnikov invariants. Several applications of this result are obtained. We show how the homotopy type of the (n+1)st homology section depends on the homotopy type of the nth homology section and the (n+1)st homology group. We prove that all homology sections of a co-H-space are co-H-spaces, all n-equivalences of the homology decomposition are...

Let G be a finite group, ${\mathbb{O}}_{G}$ the category of canonical orbits of G and $A:{\mathbb{O}}_{G}\to \mathbb{A}$b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with $Ex{t}^{n-1}(A,A\otimes A)$. Then the case $G={\mathbb{Z}}_{{p}^{k}}$ leads to an example of infinitely...

If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group ${}_{f}\subseteq G*H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism ${p}_{f}{=p|}_{f}{:}_{f}\to G$. A right inverse (section) $G{\to}_{f}$ of ${p}_{f}$ is called a coaction on G. In this paper we study ${}_{f}$ and the sections of ${p}_{f}$. We consider the following topics: the structure of ${}_{f}$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and the resulting...

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