An R-local Milnor-Moore theorem
type: Item
instance of: scholarly article
title: An R-local Milnor-Moore theorem
zbMATH Open document ID: 0684.55010
DOI: 10.1016/0001-8708(89)90016-9
author: David J. Anick
published in: Advances in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: This paper continues the program, begun by the author in J. Am. Math. Soc. 2, No.3, 417-453 (1989), of exploring an R-local form of rational homotopy theory. Over an arbitrary ring R, a functor K is defined from r- connected topological spaces to r-reduced differential graded Lie algebras. The homology of \(K(-)\) is a homotopy invariant and if \(r\geq 2\) this homology coincides with \(\pi_*(\Omega X)\otimes R\) through \(r+2\rho -4\) if \(R\ni n^{-1}\) for \(1\leq n<\rho\). The second Lie algebra, L(X), associated to a CW complex X of dimension \(\leq r\rho\) has the same homology as K(X) below dimension \(r\rho\) and satisfies \(H_*(UL(X))\cong H_*(\Omega X;R)\), where U denotes the enveloping algebra of a differential graded Lie algebra. These two isomorphisms generalize the Milnor-Moore theorem to a coefficient ring \(R\subset {\mathbb{Q}}\).
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 4121021
zbMATH Keywords: Hopf algebras
zbMATH Keywords: homotopy Lie algebras
zbMATH Keywords: homotopy invariant
zbMATH Keywords: CW complex
zbMATH Keywords: enveloping algebra
zbMATH Keywords: differential graded Lie algebra
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
full work available at URL: 0001-8708(89)90016-9
OpenAlex ID: W2050972010
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description: scientific article; zbMATH DE number 4121021