Finite subset spaces of graphs and punctured surfaces

Abstract

The $k$th finite subset space of a topological space $X$ is the space ${exp}_k\left(X\right)$ 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 ${\left({exp}_k\left(\varphi \right)\right)}_{\ast }$ induced by a map $\varphi :\Gamma \to {\Gamma }^{\prime }$ 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 $H_{\ast }\left({exp}_k\left(D_n\right)\right)$. We prove a structure theorem for this action, showing that the image of the pure braid group is nilpotent of class at most $\lfloor \left(n-1\right)∕2\rfloor$.