Volume 18, issue 2 (2018)

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Topological complexity of $n$ points on a tree

Steven Scheirer

Algebraic & Geometric Topology 18 (2018) 839–876
Abstract

The topological complexity of a path-connected space $X$, denoted by $TC\left(X\right)$, can be thought of as the minimum number of continuous rules needed to describe how to move from one point in $X$ to another. The space $X$ is often interpreted as a configuration space in some real-life context. Here, we consider the case where $X$ is the space of configurations of $n$ points on a tree $\Gamma$. We will be interested in two such configuration spaces. In the first, denoted by ${C}^{n}\left(\Gamma \right)$, the points are distinguishable, while in the second, ${UC}^{n}\left(\Gamma \right)$, the points are indistinguishable. We determine $TC\left({UC}^{n}\left(\Gamma \right)\right)$ for any tree $\Gamma$ and many values of $n$, and consequently determine $TC\left({C}^{n}\left(\Gamma \right)\right)$ for the same values of $n$ (provided the configuration spaces are path-connected).

Keywords
topological complexity, topological robotics, tree configuration spaces
Mathematical Subject Classification 2010
Primary: 57M15
Secondary: 55R80, 57Q05