Symmetric products, duality and homological dimension of configuration spaces

TL;DR

This paper investigates the homological and cohomological properties of configuration spaces of unordered points on manifolds, establishing new bounds and dualities using geometric methods and symmetric products.

Contribution

It provides new bounds on the cohomological dimension of braid spaces and introduces geometric dualities via truncated symmetric products.

Findings

Homology of braid spaces is affected by puncturing manifolds.

Established sharp connectivity bounds for symmetric products.

Refined McDuff's theorem on homological connectivity of maps.

Abstract

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'.