Directed topological complexity of spheres (Q2304012)

From the introduction: ``Topological complexity is a numerical homotopy invariant, defined by \textit{M. Farber} [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130); Topology Appl. 140, No. 2--3, 245--266 (2004; Zbl 1106.68107)] as part of his topological study of the motion planning problem from robotics. Given a path-connected space \(X\), let \(PX\) denote the space of all paths in \(X\) endowed with the compact open topology, and let \(\pi:PX \rightarrow X\times X\) denote the endpoint fibration given by \(\pi(\gamma) = (\gamma(0),\gamma (1))\). Viewing \(X\) as the configuration space of some mechanical system, one defines a motion planner on a subset \(A \subseteq X\times X\) to be a local section of \(\pi\) on \(A\), that is, a continuous map \(\sigma:A\rightarrow PX\) such that \(\pi\circ \sigma\) equals the inclusion of \(A\) into \(X\times X\). Assuming \(X\) is a Euclidean Neighbourhood Retract (ENR), the topological complexity of \(X\), denoted \(TC(X)\), is defined to be the smallest natural number \(k\) such that \(X\times X\) admits a partition into \(k\) disjoint ENRs, each of which admits a motion planner. Many basic properties of this invariant were established in the papers [op. cit.]. [...] Here we simply mention that the topological complexity of spheres was calculated in [Zbl 1038.68130]; it is given by \(TC(S^n)=2\) if \(n\) is odd and \(TC(S^n)=3\) if \(n\) is even. In the recent preprint [``Non-contractible configuration spaces'', Preprint, \url{arXiv:1709.0570}], \textit{C. A. I. Zapata} defined a variant of topological complexity for directed spaces. [...] The directed paths of a d-space form a subspace \(\overrightarrow{P}X\) of \(PX\). The endpoint fibration restricts to a map \(\chi:\overrightarrow{P}X\rightarrow PX\) which is not surjective in general. Its image, denoted \(\Gamma_X \subseteq X \times X\), is the set of \((x,y)\in X\times X\) such that there exists a directed path from \(x\) to \(y\). A \textsl{directed motion planner} on a subset \(A\subseteq \Gamma_X\) is defined to be a local section of \(\chi\) on \(A\). The directed topological complexity of the d-space \(X\), denoted \(\overrightarrow{TC}(X)\), is the smallest natural number \(k\) such that \(\Gamma_X\) admits a partition into \(k\) disjoint ENRs, each of which admits a directed motion planner. As remarked in the introduction to [loc. cit.], the directed topological complexity seems more suited to studying the motion planning problem in the presence of control constraints on the movements of the various parts of the system. It was shown in [loc. cit.] to be invariant under a suitable notion of directed homotopy equivalence, and a few simple examples were discussed. The article [\textit{E. Goubault} et al., J. Appl. Comput. Topol. 4, No. 1, 11--27 (2020; Zbl 1479.55008)] provides further examples, and proves several properties of directed topological complexity, including a product formula. It remains to find general upper and lower bounds for this invariant, and to give further computations for familiar d-spaces. The contribution of this short note is to compute the directed topological complexity of directed spheres. For each \(n\geq 1\) the directed sphere \(\overrightarrow{S^n}\) is the directed space whose underlying topological space is the boundary \(\partial I^{n+1}\) of the \((n+1)\)-dimensional unit cube, and whose directed paths are those paths which are non-decreasing in every coordinate. \textbf{Theorem}. The directed topological complexity of directed spheres is given by \[\overrightarrow{TC}(\overrightarrow{S^n})=2 \] for all \(n\geq 1\). This theorem will be proved in Sect. 3 below by exhibiting a partition of \(\Gamma_{\overrightarrow{S^n}}\) into 2 disjoint ENRs with explicit motion planners.'' For the proof of Theorem, the authors say that a d-space \((X,\overrightarrow{P}X)\) is \textsl{loop-free} if for all \(x\in X\), the fibre \(\overrightarrow{P}X(x,x)\) consists only of the constant path at \(x\). For such d-spaces the following lemma is proved: \textbf{Lemma} 2.5. Let \(X\) be a loop-free d-space for which \(\overrightarrow{TC}(X)=1\). Then for all \((x,y)\in \Gamma_X\), the corresponding fibre \(\overrightarrow{P}X(x,y)\) of the dipath space map is contractible. Then using this lemma, and the fact that \(\overrightarrow{S^n}\) is loop-free, it is proved that \(\overrightarrow{TC}(\overrightarrow{S^n})>1\). To prove that \(\overrightarrow{TC}(\overrightarrow{S^n})\leq 2\), the authors exhibit a partition \(\Gamma_{\overrightarrow{S^n}} = A_1\sqcup A_2\) into two disjoint ENRs, each equipped with a continuous directed motion planner \(\sigma_i: A_i\rightarrow \overrightarrow{P}(\overrightarrow{S^n})\). Reviewer's remark: The case \(n=1\) from the Theorem has also been proved by Goubault et al. [loc. cit.], but for the proof of the general case refer to the present paper.