Homology, Homotopy and Applications

Volume 21 (2019)

Number 2

Topological complexity of a map

Pages: 107 – 130

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n2.a7


Petar Pavešić (Faculty of Mathematics and Physics, University of Ljubljana, Slovenia)


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.


topological complexity, robotics, kinematic map, fibration, covering

2010 Mathematics Subject Classification

Primary 55M99. Secondary 68T40, 70B15.

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