A factorization of the homology of a differential graded Lie algebra

Let (L,∂) be a differential graded Lie algebra over the prime field Fp. There exists an isomorphism of Hopf algebras H*(UL)congruent withUE, where E is a graded Lie algebra (J. Pure. Appl. Algebra 83 (1992) 237–282). Suppose that L is q-reduced for some qgreater-or-equal, slanted1. We prove a generalization of a classical theorem of Sullivan (Inst. Hautes Études Sci. Publ. Math. (47) (1977) 269–331), which we use to show that there is an isomorphism of graded Lie algebras H(L,∂)congruent withE×K, where K is an abelian (qp+p−2)-reduced ideal. As a consequence, if X is a finite, q-connected, n-dimensional CW complex, and EX is its mod p homotopy Lie algebra (J. Pure. Appl. Algebra 83 (1992) 237–282), then there are isomorphisms (EX)mcongruent withπm+1(X;Fp) for mless-than-or-equals, slantmin(q+2p−3,pq−1).