#### 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.

Notes on file formats
Christopher Tuffley

Department of Mathematics, University of California at Davis

One Shields Avenue, Davis, CA 95616-8633, USA

Email: tuffley@math.ucdavis.edu

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