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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