Loop topological complexity
In order to measure the complexity of motion planning algorithms on a configuration space \(X\) of a physical system \(\mathcal{S}\), \textit{M. Farber} introduced in [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)] a homotopy invariant TC(X) called the topological complexity of \(X\). In this paper, the authors use the bi-evaluation map \(ev^{LP} : LX\rightarrow X\times X\) given by \(ev^{LP}(\gamma)=(\gamma(0),\gamma (\frac{1}{2}))\) to introduce a {loop} variant denoted \(TC^{LP}(X)\) of \(TC(X)\). Here \(LX := \{\gamma : \mathbb{S}^1\rightarrow X\}\) stands for the space of free loops in \(X\). Indeed, \(TC(X)\) is defined in terms of local sections of the path fibration \(\pi : PX\rightarrow X\times X\) while \(TC^{LP}(X)\) is given by those of \(ev^{LP}\). The main result of the paper asserts that \(TC^{LP}(X)=TC(X)\). Moreover, the authors present, in Section 3, some applications of \(TC^{LP} (-)\) and its interactions with the symmetric and monoidal topological complexities \(TC^S (-)\) and \(TC^M (-)\) introduced respectively in [\textit{M. Farber} and \textit{M. Grant}, Contemp. Math. 438, 85--104 (2007; Zbl 1143.70013)] and in [\textit{N. Iwase} and \textit{M. Sakai}, Topology Appl. 157, No. 1, 10--21 (2010; Zbl 1192.55002)].
scientific article; zbMATH DE number 6914252