Formal spaces with finite-dimensional rational homotopy

Abstract:

Let $S$ be a simply connected space. There is a certain principal fibration ${K_1} \to E \stackrel {\pi }{\to } K_0$ in which ${K_1}$ and ${K_0}$ are products of rational Eilenberg-Mac Lane spaces and a continuous map $\phi :S \to E$ such that in particular ${\phi _0} = \pi \circ \phi$ maps the primitive rational homology of $S$ isomorphically to that of ${K_0}$. A main result of this paper is the Theorem. If $\dim \pi _{\ast }(S) \otimes {\mathbf {Q}} < \infty$ then $\phi$ is a rational homotopy equivalence if and only if all the primitive homology in $H_{\ast }(S;\,{\mathbf {Q}})$ and $H_{\ast }({K_0},\,S;\,{\mathbf {Q}})$ can (up to integral multiples) be represented by spheres and disk-sphere pairs. Corollary. If $S$ is formal, $\phi$ is a rational homotopy equivalence.