Universal enveloping algebras and loop space homology. (English) Zbl 0769.57025

Halperin, Stephen

J. Pure Appl. Algebra 83, No. 3, 237-282 (1992).

If \(R\) is a principal ideal domain such that \(H_ *(\Omega X; R)\) is torsion free then \(H_ *(\Omega X; R)\) is a graded cocommutative Hopf algebra. The submodule \(P \subset H_ *(\Omega X;R)\) of primitive elements is a subgraded Lie algebra and the inclusion extends to a Hopf algebras homomorphism \[ \Phi: UP \to H_ *(\Omega X; R). \] In [Ann. Math., II. Ser. 81, 211-264 (1965; Zbl 0163.282)], J. W. Milnor and J. C. Moore show that if \(X\) is simply connected and \(R = Q\) then \(\Phi\) is an isomorphism. The first result of this paper is to extend this result to more general coefficient rings, at the cost of restricting the class of spaces \(X\). The second result consists to associate functorially to each space \(X\) in a certain class of spaces a graded Lie algebra \(L\) over \(\mathbb{F}_ p\) such that there exists a natural isomorphism of Hopf algebras \[ UL @>\cong>> H_ *(\Omega X;\mathbb{F}_ p). \] This theorem represents a considerable strengthening of a result of D. J. Anick [J. Am. Math. Soc. 2, No. 3, 417-453 (1989; Zbl 0681.55006)] which asserts that \(H_ *(\Omega X;\mathbb{F}_ p)\) is primitively generated.

Reviewer: J.C.Thomas (Villeneuve d’Ascq)

MSC:

57T25	Homology and cohomology of \(H\)-spaces
57T05	Hopf algebras (aspects of homology and homotopy of topological groups)
55P35	Loop spaces

Keywords:

Milnor-Moore theorem; \(\bmod p\) homology of loop spaces; graded cocommutative Hopf algebra; primitive elements; graded Lie algebra

Citations:

Zbl 0163.28202; Zbl 0163.282; Zbl 0681.55006

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