An R-local Milnor-Moore theorem. (English) Zbl 0684.55010

Anick, David J.

Adv. Math. 77, No. 1, 116-136 (1989).

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 \(R\subset {\mathbb{Q}}\).

Reviewer: J.C.Thomas

MSC:

55P60

Localization and completion in homotopy theory

Keywords:

Hopf algebras; homotopy Lie algebras; homotopy invariant; CW complex; enveloping algebra; differential graded Lie algebra

References:

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