## Algebraic & Geometric Topology

### 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 $expk(X)$ 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 $⌊(n−1)∕2⌋$.

#### Article information

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

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

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

#### 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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