Finite subset spaces of graphs and punctured surfaces

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, 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 [(n - 1)/2].