Topological complexity of a mapPavešić, Petar (Avtor) topological complexityroboticskinematic mapfibrationcoveringWe study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f \colon X \to Y$, 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.20192020-04-18 11:10:07Neznano115221UDK: 515.14ISSN pri članku: 1532-0073DOI: 10.4310/HHA.2019.v21.n2.a7COBISS_ID: 18590297OceCobissID: 513079065sl