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     

Nanoparticulate materials and regulatory policy in Europe: An analysis of stakeholder perspectives

The novel properties of nanoparticulate materials (NPM) and the rapid development of NPM based products have raised many unanswered questions and concerns by different stakeholders over its consequences for the environment and human health. These concerns have led to an increasing discussion in both the US and Europe about possible regulatory policies for NPM. In this article a comparative study of stakeholders’ perceptions on regulatory policy issues with NPM in Europe is presented. It was found that industry wants to regulate this area if the scientific evidence demonstrates that NPM are harmful, but also that the regulatory bodies do not find it necessary at this point of time to regulate until scientific evidence demonstrates that NPM are harmful. This research therefore shows that there will most likely not be any regulatory interventions until there is an established and convincing scientific knowledge base demonstrating that NPM can be hazardous. It is furthermore discussed in this article the different roles and responsibilities of the stakeholders in financing the research required to establish the necessary level of fundamental scientific evidence. It was also found that the activity of the regulatory bodies on this issue differ between the European countries.     

Central Pairs, Galois Theory and Automorphic Forms

A central pair over a field k of characteristic 0 consists of a finite Abelian group which is equipped with a central 2-cocycle with values in the multiplicative group k ^* of k. In this paper we use specific central pairs to construct a class of projective representations of the absolute Galois group G _k of k and if k is a number field we investigate the liftings of these projective representations to linear representations of G _k . In particular we relate these linear representations to automorphic representations. It turns out that some of these automorphic representations correspond to certain indefinite modular forms already constructed by E. Hecke.