Topological complexity has been introduced by M. Farber [Discrete Comput. Geom. 29, No. 2, 211–221 (2003; Zbl 1038.68130)] as a topological abstraction of the motion planning problem from robotics. While the notion of topological complexity has been successfully extended and applied to various configuration spaces of robots, there is little work on the relations between topological complexity and robot kinematics. In practical motion planning problems for robots manipulating certain objects, one is often not interested in the configuration of the whole robot manipulator, but only in the position of its end effector, e.g. the position of the tip of a screwdriver held by a robotic arm. The map that associates with a configuration of a robot the position of its end effector is called its work map. Instead of studying continuous local sections of the path fibration \(\pi: C^0([0,1],X) \to X \times X\), \(\pi(\gamma)=(\gamma(0),\gamma(1))\), as it is done for the definition of topological complexity, one is then interested in covering \(X \times X\) by open subsets \(U\) admitting continuous maps \(s:U \to C^0([0,1],X)\), for which \((f \circ s)(0)=f(x)\) and \((f \circ s)(1)=f(y)\) for all \((x,y) \in U\), i.e. for which \(f \circ \pi \circ s = f|_U\). If \(n+1\) is the minimal number of elements of an open cover of \(X \times X\) having that property, then the number \(\mathrm{tc}(f)=n\) is called the naive topological complexity of \(f\), as introduced in the present article. However, as \(f\) is not necessarily a fibration, \(\mathrm{tc}(f)\) does not share many of the topological features of Farber’s topological complexity. In particular, \(\mathrm{tc}(f)\) is not given as the sectional category of a fibration, making many of the typical approaches of obtaining upper and lower bounds for similarly defined numbers inaccessible.
To overcome this problem, the authors develop another notion of topological complexity of a map. Given path-connected topological spaces \(X\) and \(Y\), a a continuous map \(f: X\to Y\), the topological complexity of \(f\), denoted by \(\mathrm{TC}(f)\), is the smallest \(n \in \mathbb{N}\) for which there is an open cover \(U_0,U_1,\dots,U_n\) of \(X \times X\) such that each \(U_i\) admits a continuous \(s_i:U_i \to C^0([0,1],X)\) for which \(f \circ \pi \circ s_i\) is homotopic to \(f|_{U_i}\).
In the first section, the authors introduce the notion of \(f\)-sectional category of a map, generalizing the sectional category of a fibration. Among other statements, they show its boundedness from above by Lusternik-Schnirelmann category and establish a lower bound in terms of the cohomology ring of the underlying space. Moreover, they present alternative descriptions of \(f\)-sectional category that are analogous to the Whitehead and Ganea descriptions of Lusternik-Schnirelmann category.
For a continuous map \(f:X \to Y\), the invariant \(\mathrm{TC}(f)\) is then obtained as the \((f \times f)\)-sectional category of the path fibration \(\pi\), which is carried out in Section 2. Here, the authors draw consequences of the results of the first section and discuss the relation of \(\mathrm{TC}(f)\) to ordinary topological complexity. As their main examples, the authors discuss work maps of linkages of two links each having one degree of freedom.
In the third section, the authors introduce the rational topological complexity of a map which provides an algebraic lower bound on the topological complexity of a map in terms of Sullivan minimal models.
Section 4 introduces \(\mathrm{tc}(f)\) and discusses the relations between \(\mathrm{TC}(f)\) and \(\mathrm{tc}(f)\), as the latter number is more relevant for practically occuring work maps of robot manipulators. The authors show in Proposition 4.2 that indeed \(\mathrm{TC}(f) \leq \mathrm{tc}(f)\), so that the topologically better-behaved number \(\mathrm{TC}(f)\) may be used as a lower bound for the more practically relevant number \(\mathrm{tc}(f)\).
An alternative approach to the questions discussed in the present article was developed by P. Pavešić [Contemp. Math. 702, 61–83 (2018; Zbl 1430.70008); Homology Homotopy Appl. 21, No. 2, 107–130 (2019; Zbl 1426.55004)].