  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'.
