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

Lambe, Larry A.

doi:10.1090/S0002-9939-1986-0818472-X

Two exact sequences in rational homotopy theory relating cup products and commutators
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Proc. Amer. Math. Soc. 96 (1986), 360-364 Request permission

Abstract:

Let $X$ be an $(n - 1)$-connected topological space of finite rational type (i.e. ${H_n}(X;Q)$ 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 ${H^1}(X) \wedge {H^1}(X) \to {H^2}(X)$ 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.

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