Finite subset spaces of closed surfaces

الأصل · EN

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 show that the finite subset spaces of a connected 2-complex admit "lexicographic cell structures" based on the lexicographic order on I² 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 expₖ Sigma for each k and closed surface Sigma. In addition, we use Mayer-Vietoris arguments and the ring structure of H*(Symᵏ Sigma) to calculate the integer cohomology groups of the third finite subset space of Sigma closed and orientable.