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”.
