Two exact sequences in rational homotopy theory relating cup products and commutators

Let X be an (n 1)-connected topological space of finite rational type (i.e. H, (X; Q) is finite dimensional over Q for all n). Sullivan's notion of minimal model is used to derive two exact sequences involving the kernel of the cup product operation in dimension n and Whitehead products. The first of these generalizes both a theorem of John C. Wood [JCW] and a theorem of Dennis Sullivan [DS] and states that the kernel of the cup product map H1 ( X) A H1 ( X) H2 ( X) is rationally the dual of the second factor of the lower central series of the fundamental group. Other examples are given in the last section. 1. Statement of the theorem. Let X be an (n 1)-connected topological space and let Hn( X) denote its n th singular rational cohomology group. The object of this paper is to derive the following two exact sequences: THEOREM 1. The sequences (1.1) 0 -? (r2/r3 ? Q)* -* H1 A H1 H2 forn = 1, (1.2) 0 H2 H ((T2n-l ?9 Q)* Hn A H -u H2 for n > 1 are exact, where 7rT = 'r(X), and Fi is the ith term in the lower central series of 7rI, H' = H'(X; Q), d is dual to the commutator map [ ]: T1/F2 A 7T/F2 -r F2/F3, h is dual to the Hurewicz map X, g is dual to the composition of J'? X' and the Whitehead product [ ]: 7Tn )7T2 n -1 and A denotes exterior product if n is odd and symmetric tensor product if n is even. The sequence (1.1) above is stated for the special case of compact oriented 3-manifolds in [DS] without proof. The case F2 = 0 (i.e., 7T, abelian) of (1.1) gives Corollary 2 of [JCW]. Note that (1.1) applies to groups G and group cohomology if X = K(G; 1) is used. The author is grateful to A. Libgober for inquiring about the validity of (1.1) in its stated generality. We note that versions of these exact sequences may also be derived from known spectral sequences [BK, LS] (see also Received by the editors July 22, 1983 and, in revised form, January 10, 1985. 1980 Mathematics Subject Classification. Primary 55P62, 55Q15, 55N99.