Volume 3, issue 2 (2003)

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Finite subset spaces of graphs and punctured surfaces

Christopher Tuffley

Algebraic & Geometric Topology 3 (2003) 873–904
 arXiv: math.GT/0210315
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 $\varphi :\phantom{\rule{0.3em}{0ex}}\Gamma \to {\Gamma }^{\prime }$ 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 $⌊\left(n-1\right)∕2⌋$.

Keywords
configuration spaces, finite subset spaces, symmetric product, graphs, braid groups
Mathematical Subject Classification 2000
Primary: 54B20
Secondary: 05C10, 20F36, 55Q52