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) ≅ UE, where E is a graded Lie algebra (J. Pure. Appl. Algebra 83 (1992) 237-282). Suppose that L is q-reduced for some q ≥ 1. 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, ∂) ≅ E × K, where K is an abelian (q p + 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)m ≅ πm+1(X;Fp) for m ≤ min(q + 2 p - 3, pq - 1). © 2002 Elsevier Science B.V. All rights reserved.