Algebraic & Geometric Topology

Finite subset spaces of graphs and punctured surfaces

Christopher Tuffley

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882420

Digital Object Identifier
doi:10.2140/agt.2003.3.873

Mathematical Reviews number (MathSciNet)
MR2012957

Zentralblatt MATH identifier
1032.55014

Subjects
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

Keywords
configuration spaces finite subset spaces symmetric product graphs braid groups

Citation

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. https://projecteuclid.org/euclid.agt/1513882420


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