Finite subset spaces of graphs and punctured surfaces

Tuffley, Christopher¹

¹Department of Mathematics, University of California at Davis, One Shields Avenue, Davis, CA 95616-8633, USA

Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 873-904

Résumé

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 Γ, and study the maps (expk(ϕ))∗ induced by a map ϕ: Γ → Γ′ 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 ⌊(n − 1)∕2⌋.

Détail

DOI : 10.2140/agt.2003.3.873

Keywords: configuration spaces, finite subset spaces, symmetric product, graphs, braid groups

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