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

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 Cn(Γ), the points are distinguishable, while in the second, UCn(Γ), the points are indistinguishable. We determine TC(UCn(Γ)) for any tree Γ and many values of n, and consequently determine TC(Cn(Γ)) for the same values of n (provided the configuration spaces are path-connected).

topological complexity, topological robotics, tree configuration spaces
Mathematical Subject Classification 2010
Primary: 57M15
Secondary: 55R80, 57Q05
Received: 2 August 2016
Revised: 5 February 2017
Accepted: 4 April 2017
Published: 12 March 2018
Steven Scheirer
Department of Mathematics
Lehigh University
Bethlehem, PA
United States