Sequential parametrized topological complexity and related invariants (Q6593013)

In a series of two papers [SIAM J. Appl. Algebra Geom. 5, No. 2, 229--249 (2021; Zbl 1473.55010); Ann. Math. Artif. Intell. 90, No. 10, 999--1015 (2022; Zbl 1509.55011)], \textit{D. Cohen} et al. introduced the concept of \textit{parametrized topological complexity} \(\mathrm{TC}[p:E \to B]\) for a Hurewicz fibration \(p\). This fiberwise homotopy invariant is particularly useful in parametrized motion planning algorithms, where the goal is to move a system from a current state to a desired state within the same fiber of \(p\), ensuring continuous motion constrained to that fiber. These algorithms are designed to function under varying external conditions, offering flexibility and broad applicability.\N\N\textit{M. Farber} and \textit{A. Paul} [Topology Appl. 321, Article ID 108256, 23 p. (2022; Zbl 1510.55002)] later extended this concept by introducing the notion of \textit{sequential parametrized topological complexity}, \(\mathrm{TC}_r[p : E \to B]\), for any integer \(r \geq 2\), with the case \(r = 2\) recovering the original \(\mathrm{TC}[p : E \to B]\). This generalization requires the system to complete a sequence of tasks, each constrained to a fiber of the fibration.\N\NThe paper under review further develops this extension, focusing on the dependence of \(\mathrm{TC}_r[p : E \to B]\) on a bundle with structure group \(G\) and fiber \(X\) as a \(G\)-space. The primary aim is to establish lower and upper bounds for this invariant. To this end, the authors first show that the equivariant sequential topological complexity of \(X\) [\textit{M. Bayeh} and \textit{S. Sarkar}, J. Homotopy Relat. Struct. 15, No. 3--4, 397--416 (2020; Zbl 1459.55001); \textit{H. Colman} and \textit{M. Grant}, Algebr. Geom. Topol. 12, No. 4, 2299--2316 (2012; Zbl 1260.55007)] provides an upper bound for \(\mathrm{TC}_r[p : E \to B]\). Since this invariant is often infinite, they propose an alternative: the \textit{weak sequential equivariant topological complexity} \(\mathrm{TC}^w_{r,G}(X)\) and its variant \(\mathrm{TC}^w_{r,G}(X;P)\) with coefficients \(P\). Through examples, the authors demonstrate that these new invariants remain finite even in cases where the equivariant topological complexity is infinite.\N\NThe main result of the paper establishes the following bounds for \(\mathrm{TC}_r[p : E \to B]\):\N\N\textbf{Theorem.} Let \(p: E = X \times_G P \to B = P/G\) be a locally trivial bundle. Then the following inequalities hold: \N\[\N\mathrm{TC}^w_{r,G}(X;P) \leq \mathrm{TC}_r[p : E \to B] \leq G\text{-cat}[p : E \to B] + \mathrm{TC}^w_{r,G}(X). \N\]\NHere, \(G\text{-cat}[p : E \to B]\) is an invariant introduced by \textit{I. James} in [Topology 17, 331--348 (1978; Zbl 0408.55008)]. Several examples are provided to illustrate the applications of these results. Additionally, the authors develop a method to estimate the sectional category of towers of fibrations, which serves as a valuable technical tool for proving their main results.