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

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
Full Text:

### References:

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