Generalized Dunkl-Lipschitz Spaces

This paper deals with generalized Lipschitz spaces $\wedge^k_{\alpha,p,q}(\R)$ in the context of Dunkl harmonic analysis on $\R$, for all real $\alpha$. It also introduces a generalized Dunkl-Lipschitz spaces ${\cal T}\wedge^k_{\alpha,p,q}(\R^2_+)$ of $k$-temperature on $\R^2_+$. Some properties and continuous embedding of these spaces and the isomorphism of ${\cal T}\wedge^k_{\alpha,p,q}(\R^2_+)$ and $\wedge^k_{\alpha,p,q}(\R)$ are established

Symmetric products, duality and homological dimension of configuration spaces

We discuss various aspects of `braid spaces' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of Fred Cohen, Goryunov and Napolitano. Next we obtain a precise bound for the cohomological dimension of braid spaces. This is related to some sharp and useful connectivity bounds that we establish for the reduced symmetric products of any simplicial complex. Our methods are geometric and exploit a dual version of configuration spaces given in terms of truncated symmetric products. We finally refine and then apply a theorem of McDuff on the homological connectivity of a map from braid spaces to some spaces of `vector fields'.Comment: This is the version published by Geometry & Topology Monographs on 26 July 2008. arXiv-admin note: this is the same article as the author's version arXiv:math/061143