Topological complexity of a map

Petar Pavešić

Homology, homotopy, and applications, Tome 21 (2019) no. 2, pp. 107-130.

Résumé

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.

Détail

DOI : 10.4310/HHA.2019.v21.n2.a7

Classification : 55M99, 68T40, 70B15
Keywords: topological complexity, robotics, kinematic map, fibration, covering