Geodesic complexity for non-geodesic spaces (Q2070491)

\textit{M. Farber} defined the notion of topological complexity, \(TC(X)\), of an arbitrary topological space [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)]. \(TC(X)\) is defined to be the smallest natural number \(k\) such that \(X\times X\) admits a partition into \(k\) disjoint ENRs, each of wich admits a ``motion planer''. Then, \textit{D. Recio-Mitter} defined, in [J. Appl. Comput. Topol. 5, No. 1, 141--178 (2021; Zbl 1485.53050)], the notion of geodesic complexity \(GC(X)\) of a metric space \(X\) to be the smallest number \(k\) such that \(X \times X\) can be partitioned into \(k + 1\) locally compact sets \(E_i\), \(0 \leq i \leq k\), such that on each \(E_i\), there is a continuous map \(s_i : E_i \rightarrow PX \), called a geodesic motion planning rule (GMPR) on \(E_i\), such that, for all \((x_0,x_1)\in E_i\); \(s_i(x_0,x_1)\in E_i\), \(s_i(x_0,x_1)\) is a (minimal) geodesic from \(x_0\) to \(x_1\). This is an analogue of Farber's notion of topological complexity \(TC(X)\) and for a metric space \(X\), \(TC(X)\leq GC(X)\). ``These notions are of particular interest if \(X\) is a space of configurations of one or more robots'' (from the introduction). A geodesic space is one in which for all pairs \((x_0,x_1)\) of points, there is a geodesic from \(x_0\) to \(x_1\). If \(X\) is not geodesic, then by definition \(GC(X)=\infty\). But in two other articles, \textit{D. M. Davis} [``Geodesics in the configuration spaces of two points in $\mathbb{R}^n$'', Tbilisi Math. J. 14, 149--162 (2021)] and \textit{D. M. Davis} et al. [``Two robots moving geodesically on a tree'', Alg. Geom. Topol. (to appear)], some non-geodesic spaces \(X\) were replaced by homotopically equivalent geodesic spaces, whose \(GC\) was computed and interpreted as representing \(GC(X)\) (this is reasonable, since \(TC\) is a homotopy invariant). Now, let \(F(X, 2)\) denote the space of ordered pairs of distinct points of \(X\), with the metric induced from \(X \times X\). In the paper from 2021 [loc. cit.], the author replaced the non-geodesic space \(F(\mathbb{R}^n,2)\) by the homotopically equivalent geodesic space \(F_\epsilon(\mathbb{R}^n,2)\) consisting of points \((x_0,x_1)\) satisfying \(d(x_0,x_1)<\epsilon\). But the author writes: ``We determined explicit geodesics in \(F_\epsilon(\mathbb{R}^n,2)\), but the work was quite complicated''. Then, in the present paper the author proceeds as follows: For a topological space \(Y\), \(P(Y)=Y^I\) denotes the path space of continuous maps \(I \rightarrow Y\), with the compact-open topology, and \(P(Y;y_0,y_1)\) ) the subspace consisting of paths from \(y_0\) to \(y_1\), and the following definitions: \textbf{Definition} 1.1. Let \(X\) be a metric space whose completion \(\overline{X}\) is geodesic. For \(x_0, x_1 \in X\), a \textit{near-geodesic} from \(x_0\) to \(x_1\) is a continuous map \(\phi:I\rightarrow P(\overline{X};x_0,x_1)\) satisfying: i. \(\phi(0)\) is geodesic in \(\overline{X}\) from \(x_0\) to \(x_1\); ii. \(\phi((0,1])\subset P(X;x_0,x_1)\); iii. length \(\phi(s)\) is finite for all \(s\) and if \(s_n\rightarrow 0\), then \(length(\phi(s_n))\rightarrow lenght (\phi(0))\). ``The length of a path is defined (as in [Recio-Mitter, loc. cit.]) to be the supremum of sums of distances between successive points on the path. Item (iii) of the above definition guarantees that for values of \(s\) close enough to \(0\), the paths \(\phi(s)\) in \(X\) are good approximations to the geodesic in \(\overline{X}\). If there is a geodesic in \(X\) from \(x_0\) to \(x_1\), one can use the constant homotopy, but we still call it a near-geodesic for uniformity''. \textbf{Definition} 1.2 For \(E\subset X\times X\), a \textit{near-geodesic motion planning rule} (NGMPR) on \(E\) is a continuous map \(\Phi\) from \(E\) to \(P(\overline{X})^I \) such that, for all \((x_0,x_1)\in E, \Phi(x_0,x_1)\) is a near-geodesic from \(x_0\) to \(x_1\), with the additional proviso that \(\Phi(x_0,x_1)(0)\) is a geodesic in \(X\) if one exists. The \textsl{geodesic complexity} \(GC(X)\) is defined as the smallest \(k\), such that \(X\times X\) can be partitioned into locally compact sets \(E_0,\dots,E_k\), such that each \(E_i\) has an NGMPR. \textbf{Proposition} 1.3 If \(X\) is geodesic, then the definition of \(GC(X)\) in Definition 1.2 agrees with that given by Recio-Mitter [loc. cit.]. Then, as with Recio-Mitter, the inequality \(TC(X)\leq GC(X)\) is true, since if \(\Phi\) is an NGMPR on \(E\), then the map \(E\rightarrow P(X)\) defined by \((x_0,x_1)\rightarrow \Phi(x_0,x_1)(\frac{1}{2})\) is a motion planning rule on \(E\). But, in Section 2 of the paper, the author proves that \(GC(X)=TC(X)\) for the following non-geodesic spaces \(X\), by constructing explicit NGMPRs: \(\bullet\) \(\mathbb{R}^n \setminus Q\), with \(n\geq 2\) and \(Q\) a finite subset, \(\bullet\) \(F(\mathbb{R}^n,2)\), with \(n\geq 2\), \(\bullet\) \(F(\mathbb{R}^n -\{x_0\},2)\), with \(n\geq 2\), \(\bullet\) the unordered configuration space \(C(\mathbb{R}^2-\{x_0\},2)\), \(\bullet\) \(F(Y,2)\), where \(Y\) is a graph with exactly one essential vertex, of order 3. But in Section 3, the author studies \(GC(X)\) for \(X=F(\mathbb{R}^n-Q,2)\) if \(Q\) is a finite subset with at least two points, after which the author writes ``this might be a case in which \(GC(X) >TC(X)\)''. In fact, he proves (Theorem 3.1) that for \(2\leq |Q|<\infty\) and \(X=F(\mathbb{R}^n-Q,2)\), \(GC(X)\leq 5\). Using a result of \textit{M. Farber} et al. [Contemp. Math. 438, 75--83 (2007; Zbl 1143.55300)], namely that \(TC(\mathbb{R}^n-Q,2)=4\) the author writes: ``We think it is likely that, at least for \(n=2\), our result for \(GC(X)\) is sharp, which would give a nice example of \(TC<GC\)''. In the proof of Theorem 3.1, the author obtains at first a partition of \(X\) into 18 subsets, on each of which there is a GMPR or NGMPR, and then these are grouped into six collections of topologically disjoint subsets. Section 4, the last of the article, is entitled ``Configuration spaces of graphs''. For a metric space \((X,d)\), the \textsl{intrinsic metric} \(d_I(x,y)\) on the set \(X\) is defined as the infimum of the d-lengths of paths in \(X\) from \(x\) to \(y\). Then \(F_I(Y,2)\) denote \(F(Y,2)\) in the intrinsic metric.With this notation the main result of Section 4 is the following. \textbf{Theorem} 4.4 If \(Y\) denotes the \(Y\) graph, then \(GC(F_I(Y,2))=TC(F(Y,2))=1\).