Finite subset spaces of graphs and punctured surfaces

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 $Γ$ , and study the maps ${({exp}_{k} (ϕ))}_{*}$ induced by a map $ϕ : Γ \to Γ^{'}$ 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 $⌊ (n - 1) ∕ 2 ⌋$ .

Keywords

configuration spaces, finite subset spaces, symmetric product, graphs, braid groups

Mathematical Subject Classification 2000

Primary: 54B20

Secondary: 05C10, 20F36, 55Q52

References

Forward citations

Publication

Received: 21 February 2003

Revised: 16 September 2003

Accepted: 23 September 2003

Published: 25 September 2003

Authors

Christopher Tuffley
Department of Mathematics University of California at Davis One Shields Avenue Davis, CA 95616-8633 USA

Volume 3, issue 2 (2003)

Christopher Tuffley