zbMATH — the first resource for mathematics

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

55P60 Localization and completion in homotopy theory
Full Text: DOI
[1] Anick, D.J, Homotopy exponents for spaces of cateory two, (), 24-52
[2] Anick, D.J, Hopf algebras up to homotopy, J. amer. math. soc., 2, No. 3, (1989) · Zbl 0681.55006
[3] Aubry, M; Lemaire, J.-M, Homotopies d’algèbres de Lie et de leurs algèbres enveloppantes, (), 26-30
[4] Cohen, F.R; Moore, J.C; Neisendorfer, J, Torsion in homotopy groups, Ann. of math., 109, 121-168, (1979) · Zbl 0405.55018
[5] Eilenberg, S; Moore, J.C, Homology and fibrations I, Comment. math. helv., 40, 199-236, (1966) · Zbl 0148.43203
[6] James, I.M, Reduced product spaces, Ann. of math., 62, 170-197, (1955) · Zbl 0064.41505
[7] Lemaire, J.-M, Algèbres connexes et homologie des espaces de lacets, () · Zbl 0293.55004
[8] Milnor, J; Moore, J.C, On the structure of Hopf algebras, Ann. of math., 81, 211-264, (1965) · Zbl 0163.28202
[9] Neisendorfer, J, Primary homotopy theory, Memoirs amer. math. soc., Vol. 25, No. 232, (1980) · Zbl 0446.55002
[10] Quillen, D.G, Rational homotopy theory, Ann. of math., 90, 205-295, (1969) · Zbl 0191.53702
[11] Spanier, E, Algebraic topology, () · Zbl 0810.55001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.