Finite subset spaces of closed surfaces

The kth finite subset space of a topological space X is the space expkX of non-empty finite subsets of X of size at most k , topologised as a quotient of X . The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X . We show that the finite subset spaces of a connected 2–complex admit “lexicographic cell structures” based on the lexicographic order on I and use these to study the finite subset spaces of closed surfaces. We completely calculate the rational homology of the finite subset spaces of the two-sphere, and determine the top integral homology groups of expkΣ for each k and closed surface Σ. In addition, we use Mayer-Vietoris arguments and the ring structure of H∗(SymΣ) to calculate the integer cohomology groups of the third finite subset space of Σ closed and orientable. AMS Classification 55R80 (54B20 55Q52)