Sadok Kallel
Published 2006-11-14, updated 2008-07-06Version 2
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''.
Comments: This is a major revision. We fill in some incomplete arguments of proof. Old theorem 1.8 (k=1 case) corrected. Improved hypothesis in theorem 1.1. Some sections have been rearranged. Finally a title change. To appear in Geometry and Topology Monographs
Symmetric products, duality and homological dimension of configuration spaces
Computational topology for configuration spaces of hard disks
On the Homology of Configuration Spaces $C((M,M_o)\times {\bold R}^n; X)$
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