Topological complexity of $n$ points on a tree

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 $\mathrm{\Gamma }$. We will be interested in two such configuration spaces. In the first, denoted by $C^n\left(\mathrm{\Gamma }\right)$, the points are distinguishable, while in the second, ${UC}^n\left(\mathrm{\Gamma }\right)$, the points are indistinguishable. We determine $TC\left({UC}^n\left(\mathrm{\Gamma }\right)\right)$ for any tree $\mathrm{\Gamma }$ and many values of $n$, and consequently determine $TC\left(C^n\left(\mathrm{\Gamma }\right)\right)$ for the same values of $n$ (provided the configuration spaces are path-connected).