Topological complexity of n points on a tree

Scheirer, Steven

doi:10.2140/agt.2018.18.839

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Topological complexity of $n$ points on a tree

Steven Scheirer

Algebraic & Geometric Topology 18 (2018) 839–876

DOI: 10.2140/agt.2018.18.839

Abstract

The topological complexity of a path-connected space $X$ , denoted by $TC (X)$ , 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 $Γ$ . We will be interested in two such configuration spaces. In the first, denoted by $C^{n} (Γ)$ , the points are distinguishable, while in the second, ${UC}^{n} (Γ)$ , the points are indistinguishable. We determine $TC ({UC}^{n} (Γ))$ for any tree $Γ$ and many values of $n$ , and consequently determine $TC (C^{n} (Γ))$ 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

References

Publication

Received: 2 August 2016

Revised: 5 February 2017

Accepted: 4 April 2017

Published: 12 March 2018

Authors

Steven Scheirer
Department of Mathematics Lehigh University Bethlehem, PA United States
https://www.lehigh.edu/~sts413/

Volume 18, issue 2 (2018)