Finite subset spaces of graphs and surfaces.

摘要

The kth finite subset space of a topological space X is the space expkX of non-empty subsets of X of size at most k, topologised as a quotient of Xk. The construction is a homotopy functor, and may be thought of as a union over 1 ≤ j ≤ k of configuration spaces of unordered j-tuples of distinct points in X.; We study finite subset spaces in the context of the circle, graphs, surfaces, and maps between these spaces. We show that expkS 1 has the homotopy type of an odd dimensional sphere of dimension k or k - 1, that the inclusion map exp2ℓ-1S1 ≃ S2ℓ-1 ↪ exp2ℓS1 ≃ S2ℓ-1 has degree two, and that exp kS1expk-2 S1 has the homotopy type of a ( k - 1, k)-torus knot complement. These results generalise known facts that exp2S1 is a Mobius strip with boundary exp1S 1, and that exp3S1 is the three-sphere with exp1S1 forming a trefoil knot inside it.; We build cell structures for the finite subset spaces of a connected graph Gamma and use these to calculate H*(exp kGamma) and give recipes for the maps induced on homology by a map of graphs &phis;: Gamma → Gamma'. These recipes take the form of a ring structure on the cellular chain groups with respect to which chain maps (expk&phis;) ♯ are ring homomorphisms. The results apply to punctured surfaces also, by homotopy equivalence, and we use them to prove a structure theorem for the induced action of the braid group Bn, acting as the mapping class group of a punctured disc.; We show that the finite subset spaces of a connected finite 2-complex admit "lexicographic cell structures" based on the lexicographic order on I2, and use these to calculate the rational homology of expkS2 and the top integral homology groups of expkSigma for each k and closed surface Sigma. In addition, we use the Mayer-Vietoris sequence and the ring structure of H*(Sym kSigma) to completely calculate the cohomology groups of exp3Sigma for Sigma closed and orientable.; Finally, we use the homology of expkGamma to prove two vanishing theorems for the homotopy and homology groups of the finite subset spaces of a connected cell complex.