Splitting off rational parts in homotopy types

Abstract

It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (see Theorem 21.3 of [L. Fuchs, Infinite Abelian Groups, vol. I, Academic Press, New York–London, 1970]). In this paper, conditions to split off rational parts in homotopy types from a given space are studied in terms of a variant of Hurewicz map, say $\overline{\rho }:[S_{\mathbb{Q}}^n,X]\to H_n(X;\mathbb{Z})$ and generalised Gottlieb groups. This yields decomposition theorems on rational homotopy types of Hopf spaces, T-spaces and Gottlieb spaces, which has been known in various situations, especially for spaces with finiteness conditions.