Topological complexity of a map

We study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f : X \to Y$, where $f$ can be a kinematic map from the configuration space $X$ to the working space $Y$ of a robot arm or a similar mechanism. Then one can associate to $f$ a number $\mathrm{TC}(f)$, which is, roughly speaking, the minimal number of continuous rules that are necessary to construct a complete manipulation algorithm for the device. Examples show that $\mathrm{TC}(f)$ is very sensitive to small perturbations of $f$ and that its value depends heavily on the singularities of $f$. This fact considerably complicates the computations, so we focus here on estimates of $\mathrm{TC}(f)$ that can be expressed in terms of homotopy invariants of spaces $X$ and $Y$, or that are valid if $f$ satisfies some additional assumptions like, for example, being a fibration.

Some of the main results are the derivation of a general upper bound for $\mathrm{TC}(f)$, invariance of $\mathrm{TC}(f)$ with respect to deformations of the domain and codomain, proof that $\mathrm{TC}(f)$ is a FHE invariant, and the description of a cohomological lower bound for $\mathrm{TC}(f)$. Furthermore, if $f$ is a fibration we derive more precise estimates for $\mathrm{TC}(f)$ in terms of the Lusternik–Schnirelmann category and the topological complexity of $X$ and $Y$. We also obtain some results for the important special case of covering projections.

Keywords

topological complexity, robotics, kinematic map, fibration, covering

2010 Mathematics Subject Classification

Primary 55M99. Secondary 68T40, 70B15.

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The author was supported by the Slovenian Research Agency research grant P1-0292 and research project J1-7025.

Received 31 August 2018

Received revised 26 October 2018

Accepted 31 October 2018

Published 13 February 2019