Finite subset spaces of graphs and punctured surfaces

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The kth finite subset space of a topological space X is the space expₖ X of non-empty finite subsets of X of size at most k, topologised as a quotient of Xᵏ. 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 (expₖ 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 Bₙ may be regarded as the mapping class group of an n-punctured disc Dₙ, and as such it acts on H*(expₖ Dₙ). 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).