Bidirectional sequential motion planning (Q2024058)

The topological complexity of a space \(X,\) denoted as \(\mbox{TC}(X)\), is a homotopy invariant introduced by \textit{M. Farber} in [Discrete Comput. Geom. 29 No. 2, 211--221 (2003; Zbl 1038.68130)] in order to study the motion planning problem in robotics from a topological perspective. It is defined as the sectional category of the bi-evaluation fibration \(e:X^I\rightarrow X\times X\), \(\alpha \mapsto (\alpha (0),\alpha (1))\) (here \(X^I\) denotes the free path space of \(X\) equipped with the compact-open topology). This notion can be extended by considering the sectional category of the multi-evaluation fibration \(e:X^I\rightarrow X^n,\) \(\alpha \mapsto (\alpha (0),\alpha (\frac{1}{n-1}),\dots,\alpha (\frac{n-2}{n-1}),\alpha (1))\) obtaining this way the higher analogues \(\mbox{TC}_n(X)\). Obviously, we have \(\mbox{TC}(X)=\mbox{TC}_2(X)\) as a particular case. A special symmetric version of \(\mbox{TC}_n(-)\), called \(n\)-th symmetrized topological complexity and denoted by \(\mbox{TC}^{\Sigma }_n(-)\) was defined in [\textit{I. Basabe} et al., Algebr. Geom. Topol. 14, No. 4, 2103--2124 (2014; Zbl 1348.55005)]. This notion turns out to be a homotopy invariant. However, for \(n>2\) it is not clear whether it is related to the motion planning problem. In the paper under review the author introduces a new symmetric version of \(\mbox{TC}_n(-)\), a homotopy invariant called \(n\)-th bidirectional topological complexity and denoted by \(\mbox{TC}^{\beta }_n(-)\). It is closely related to \(\mbox{TC}^{\Sigma }_n(-)\) and also to the motion planning problem. Moreover, we have a chain of inequalities \[\mbox{TC}_n(X)\leq \mbox{TC}^{\beta }_n(X)\leq \mbox{TC}^{\Sigma }_n(X)\] for all \(n\geq 2,\) and \(\mbox{TC}^{\beta }_2(X)=\mbox{TC}^{\Sigma }_2(X)\). After giving some interesting properties of bidirectional sectional category, where we can mention cohomological lower bounds and other useful bounds, the author gives some calculations concerning the \(m\)-sphere, namely \[ \mbox{TC}^{\beta }_n(S^m)=\begin{cases} n+1, & \mbox{if}\hspace{4pt}m\hspace{4pt}\mbox{is}\hspace{4pt}\mbox{even} \\ n, & \mbox{if both}\hspace{4pt}m,n\hspace{4pt}\mbox{are}\hspace{4pt}\mbox{odd} \end{cases} \] This fact together with the techniques developed in the paper lead to the computation of \(\mbox{TC}^{\Sigma }_n(S^m)\) and \(\mbox{TC}^{\Sigma }_n(\mathbb{R}\mbox{P}^m)\), for special cases of \(n\) and \(m.\) In order to illustrate the relationship to the motion planning problem, in the final section of the paper the author gives a description of specific motion planners that realize the computation \(\mbox{TC}^{\beta }_2(S^m)=\mbox{TC}^{\Sigma }_2(S^m)=3.\)