A Torsion-Free Milnor-Moore Theorem
Jonathan A. Scott
Abstract
Let ΩX be the space of Moore loops on a finite, q-connected, n-dimensional CW complex X, and let R be a subring of Q containing 1/2. Let p(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M / Torsion M. We show that the inclusion of the sub-Lie algebra P of primitive elements of FH*(ΩX;R) induces an isomorphism of Hopf algebras UP = FH*(ΩX;R), provided p(R) > n/q - 1. Furthermore, the Hurewicz homomorphism induces an embedding of F(π*(ΩX) R) in P, with torsion cokernel. As a corollary, if X is elliptic, then FH*(ΩX;R) is a finitely-generated R-algebra.
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