Algebraic & Geometric Topology

Finite subset spaces of graphs and punctured surfaces

Christopher Tuffley

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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:ΓΓ 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 (n1)2.

Article information

Algebr. Geom. Topol., Volume 3, Number 2 (2003), 873-904.

Received: 21 February 2003
Revised: 16 September 2003
Accepted: 23 September 2003
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54B20: Hyperspaces
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 20F36: Braid groups; Artin groups 55Q52: Homotopy groups of special spaces

configuration spaces finite subset spaces symmetric product graphs braid groups


Tuffley, Christopher. Finite subset spaces of graphs and punctured surfaces. Algebr. Geom. Topol. 3 (2003), no. 2, 873--904. doi:10.2140/agt.2003.3.873.

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