The milnor-moore theorem in tame homotopy theory

LetX be a 1-connected space with Moore loop space ΩX. By a well-known theorem of J. W. Milnor and J. C. Moore [7] the Hurewicz homomorphism induces an isomorphism of Hopf algebrasU(π_*(ΩX) ⊗Q)→H _*(ΩX;Q). HereU(−) denotes the universal enveloping algebra and the Lie bracket on π_*(ΩX) ⊗Q is given by the Samelson product. Assume now thatX is the geometric realization of anr-reduced simplicial set,r≥3. LetL _X be a differential graded free Lie algebra over ℤ describing the tame homotopy type ofX according to the theory of [4]. Then the main result of the present paper is the construction of a sequence of morphisms of differential graded algebras betwenU(L _X ) and the algebraC U _*(ΩX)z of normalized cubical chains on ΩX such that the induced morphisms on homology with coefficientsR _k are isomorphismsH _r-1+l (U(L _x );R _k ) ≅H _r-1+l C U _*(ΩX);R _k ) forl≤k; hereR ₀⊆R ₁⊆… is a tame ring system, i. e.R _k )⊑Q and each primep with 2p−3≤k is invertible inR _k . However, it is no longer true that the Pontrjagin algebraH _≤r−1+k (ΩX; R _k ) of ΩX in degrees ≤r−1+k is determined by π_*(ΩX) or by a cofibrant (-fibrant) modelM of π_*(ΩX) as will be shown by an example. But there is a filtration onH _≤r−1+k (ΩX; R _k ) such that the associated graded algebra is isomorphic toH _≤r−1+k (U(M); R _k ).This will be proved by using a filtered Lie algebra model ofX constructed from a bigraded model of π_*(ΩX). Supported by a CNRS grant and PROCOPE Supported by PROCOPE