The th finite subset space of a topological space is the space of non-empty finite subsets of of size at most , topologised as a quotient of . The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in . We calculate the homology of the finite subset spaces of a connected graph , and study the maps induced by a map between two such graphs. By homotopy functoriality the results apply to punctured surfaces also. The braid group may be regarded as the mapping class group of an –punctured disc , and as such it acts on . We prove a structure theorem for this action, showing that the image of the pure braid group is nilpotent of class at most .
"Finite subset spaces of graphs and punctured surfaces." Algebr. Geom. Topol. 3 (2) 873 - 904, 2003. https://doi.org/10.2140/agt.2003.3.873