In the present article, the authors modify methods from their earlier article [C. A. Ipanaque Zapata and J. González, Discrete Math. Algorithms Appl. 12, No. 3, Article ID 2050040, 19 p. (2020; Zbl 1465.55004)] to carry out a construction of sequential motion planning algorithms for configuration spaces of \(k\) points in \(\mathbb{R}^d \setminus Q_r\), where \(d \geq 2\) and \(Q_r \subset \mathbb{R}^d\) is a subset with \(r\) elements. These spaces are explicitly given as \(F(\mathbb{R}^d\setminus Q_r,k) = \{ (x_1,\dots,x_k)\in (\mathbb{R}^d\setminus Q_r)^k \ | \ x_i \neq x_j \text{ if } i \neq j\}\) and appear as fibers in Fadell-Neuwirth fibrations of configuration spaces. The thus-obtained motion planning algorithms have the minimal number of domains of continuity that is prescribed by their higher topological complexities, see [I. Basabe et al., Algebr. Geom. Topol. 14, No. 4, 2103–2124 (2014; Zbl 1348.55005)] for details on sequential motion planning algorithms.
While the higher topological complexities of \(F(\mathbb{R}^d\setminus Q_r,k)\) have been computed by the second author and M. Grant [Proc. Am. Math. Soc. 143, No. 10, 4503–4512 (2015; Zbl 1325.55008)], the methods used therein are based on homotopy theory and in particular do not yield explicit motion planning algorithms.
After an introductory first section, the authors recall well-known results about the homotopy invariance of higher topological complexities in Section 2. In that section they further give explicit formulas for how sequential motion planners can be carried over to another space by a deformation or a dominating map. In Section 3, the authors construct motion planning algorithms on \(F(\mathbb{R}^d\setminus Q_r,k)\) for \(d \geq 2\), \(k \geq 2\) and \(r \geq 2\) that are optimal in the sense of Farber’s topological complexity, i.e. that have \(\mathrm{TC}(F(\mathbb{R}^d\setminus Q_r,k))\) distinct domains of continuity. The construction is extended in Section 4 to obtain sequential motion planning algorithms having \(\mathrm{TC}_m(F(\mathbb{R}^d\setminus Q_r,k))\) domains of continuity. Here, by \(\mathrm{TC}_m\) we denote Rudyak’s higher topological complexities, see again [I. Basabe et al., loc. cit.] for details.
Methodically, the authors combine explicit and elementary descriptions of such algorithms on special subsets of the \(F(\mathbb{R}^d\setminus Q_r,k)\) with deformation results for configuration spaces from their earlier work [op. cit.].