  The kth finite subset space of a topological space X is the space exp_k X of
non-empty finite subsets of X of size at most k, topologised as a quotient of
X^k. The construction is a homotopy functor and may be regarded as a union of
configuration spaces of distinct unordered points in X. We calculate the
homology of the finite subset spaces of a connected graph Gamma, and study the
maps (exp_k phi)_* induced by a map phi:Gamma -> Gamma' between two such
graphs. By homotopy functoriality the results apply to punctured surfaces also.
The braid group B_n may be regarded as the mapping class group of an
n-punctured disc D_n, and as such it acts on H_*(exp_k D_n). We prove a
structure theorem for this action, showing that the image of the pure braid
group is nilpotent of class at most floor((n-1)/2).
