Algebraic and Geometric Topology 3 (2003), paper no. 29, pages 873-904.

Finite subset spaces of graphs and punctured surfaces

Christopher Tuffley

Abstract. The k-th finite subset space of a topological space X is the space exp_k(X) of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. 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 Gamma, and study the maps (exp_k(phi))_* induced by a map phi: Gamma --> Gamma' between two such graphs. By homotopy functoriality the results apply to punctured surfaces also. The braid group B_n may be regarded as the mapping class group of an n-punctured disc D_n, and as such it acts on H_*(exp_k(D_n)). We prove a structure theorem for this action, showing that the image of the pure braid group is nilpotent of class at most floor((n-1)/2).

Keywords. Configuration spaces, finite subset spaces, symmetric product, graphs, braid groups

AMS subject classification. Primary: 54B20. Secondary: 05C10, 20F36, 55Q52.

DOI: 10.2140/agt.2003.3.873

E-print: arXiv:math.GT/0210315

Submitted: 21 February 2003. (Revised: 16 September 2003.) Accepted: 23 September 2003. Published: 25 September 2003.

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

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