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Let S be a simply connected space. There is a certain principal fibration K1 E -Ko in which K1 and Ko are products of rational Eilenberg-Mac Lane spaces and a continuous map 4: S E such that in particular 00 = 4T o 4 maps the primitive rational homology of S isomorphically to that of Ko. A main result of this paper is the THEOREM. If dim sro(S) 0 Q < oo then 4) is a rational homotopy equivalence if and only if all the primitive homology in H*(S; Q) and H*(KO, S; Q) can (up to integral multiples) be represented by spheres and disk-sphere pairs. COROLLARY. If S is formal, 4 is a rational homotopy equivalence.

Formal spaces with finite-dimensional rational homotopy