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


Author: Larry A. Lambe
Journal: Proc. Amer. Math. Soc. 96 (1986), 360-364
MSC: Primary 55P62; Secondary 55N99, 55Q15
DOI: https://doi.org/10.1090/S0002-9939-1986-0818472-X
MathSciNet review: 818472
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0818472-X
Keywords: de Rham complex, Hurewicz map, lower central series, $ 1$-minimal model, minimal model
Article copyright: © Copyright 1986 American Mathematical Society

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