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Symmetric products, duality and homological dimension of configuration spaces

Sadok Kallel

Geometry & Topology Monographs 13 (2008) 499–527

DOI: 10.2140/gtm.2008.13.499

arXiv: 0904.1024


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”.

To Fred Cohen on his 60th birthday


braid, configuration space, cohomological dimension, symmetric product, duality, connectivity bounds

Mathematical Subject Classification

Primary: 55R80

Secondary: 18G20, 55S15


Received: 1 July 2006
Revised: 31 May 2008
Accepted: 16 June 2008
Published: 26 July 2008

Sadok Kallel
Université des Sciences et Technologies de Lille
Laboratoire Painlevé, U.F.R de Mathématiques
59655 Villeneuve d'Ascq