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Finite subset spaces of graphs and punctured surfaces

Christopher Tuffley

Algebraic & Geometric Topology 3 (2003) 873–904

arXiv: math.GT/0210315


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.

configuration spaces, finite subset spaces, symmetric product, graphs, braid groups
Mathematical Subject Classification 2000
Primary: 54B20
Secondary: 05C10, 20F36, 55Q52
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Received: 21 February 2003
Revised: 16 September 2003
Accepted: 23 September 2003
Published: 25 September 2003
Christopher Tuffley
Department of Mathematics
University of California at Davis
One Shields Avenue
Davis, CA 95616-8633