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A TORSION-FREE MILNOR–MOORE THEOREM

Published online by Cambridge University Press:  20 May 2003

JONATHAN A. SCOTT
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE [email protected]
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Abstract

Let $\Omega X$ be the space of Moore loops on a finite, $q$-connected, $n$-dimensional CW complex $X$, and let $R\subset\Q$ be a subring containing 1/2. Let $\rho\left(R\right)$ be the least non-invertible prime in $R$. For a graded $R$-module $M$ of finite type, let $FM = M / {\rm Torsion}\,M$. We show that the inclusion $P \subset FH_{*}\left(\Omega X;R\right)$ of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras $$UP \overset{\cong}{\longrightarrow} FH_{*}\left(\Omega X;R\right)},$$ provided that $\rho\left(R\right) \geqslant n/q$. Furthermore, the Hurewicz homomorphism induces an embedding of $F(\pi_{*}\left(\Omega X\right)\otimes R)$ in $P$, with $P/F(\pi_{*}\left(\Omega X\right)\otimes R)$ torsion. As a corollary, if $X$ is elliptic, then $FH_{*}\left(\Omega X;R\right)$ is a finitely generated $R$-algebra.

Type
Research Article
Copyright
The London Mathematical Society 2003

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