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

Abstract

Let X X be an ( n − 1 ) (n - 1) -connected topological space of finite rational type (i.e. H n ( X ; Q ) {H_n}(X;Q) is finite dimensional over Q Q for all n 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 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 H 1 ( X ) ∧ H 1 ( X ) → H 2 ( X ) {H^1}(X) \wedge {H^1}(X) \to {H^2}(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.