Sequential parametrized motion planning and its complexity
Let \(p:E\to B\) be a Hurewicz fibration and \(E^I_B\) is a subspace of \(E^I\) consisting of all paths \(\alpha: I\to E\) such that \(p\alpha:I\to B\) is constant. Fix an integer \(r \geq 2\) and put \[ E_B^r=\{(e_1,\dots,e_r)\in E^r; p(e_1)=\dots =p(e_r)\}. \] Take \(r\) points \(0\leq t_1<\dots<t_r\leq 1\) in \(I\). Consider the fibration \[ \Pi_r: E_B^I\to E_B^r,\quad \Pi_r(\alpha)=(\alpha(t_1, \dots, \alpha(t_r)). \] The \(r\)-th sequential parametrized topological complexity of the fibration \(p : E\to B\), denoted \(\operatorname{TC}_r[p]:=\operatorname{TC}_r[p : E\to B]\), is defined as the sectional category of the fibration \(\Pi_r\). If \(B\) is the one-point space then \(\operatorname{TC}_r[p]\) is equal to the sequential topological complexity \(\operatorname{TC}_r(E)\) described in [\textit{I. Basabe} et al., Algebr. Geom. Topol. 14, No. 4, 2103--2124 (2014; Zbl 1348.55005); \textit{Y. B. Rudyak}, Topology Appl. 157, No. 5, 916--920 (2010; Zbl 1187.55001)]. So, in general situation, the space \(B\) turns out to be a space of parameters. This justified the name ``parametrizes'' in the definition. The authors settles some properties of \(\operatorname{TC}_r[p]\) (fiber homotopy invariance, behaviour with respect to products, an inequality \(\operatorname{TC}_{r+1}[p]\geq \operatorname{TC}_r[p]\), upper and lover bounds, cup-lengths estimations).Then they treat in details the sequential parametrized topological complexity of the Fadell-Neuwirth bundle. In the last section the authors consider the \(\operatorname{TC}\)-generation \(\mathscr T(t)_{[p]}=\sum_r\operatorname{TC}_r[p]\cdot t^r\) associated with the fibration \(p:E \to B\). The authors give some examples and pose the following interesting question: When the generating \(\operatorname{TC}_{[p]}\) is a rational function or, more specifically, when the sequence \(\{\operatorname{TC}r[p]\}\) is eventually constant. For example, this holds for Fadell-Neuwirth bundles.
scientific article; zbMATH DE number 7605020
scientific article