Sequential parametrized topological complexity of sphere bundles (Q6633076)

This article investigates the sequential parametrized topological complexity of sphere bundles, extending prior research on parametrized motion planning algorithms, such as those studied by the authors in [Topology Appl. 321, Article ID 108256, 23 p. (2022; Zbl 1510.55002)] and by the first author and \textit{S. Weinberger} in [Topol. Methods Nonlinear Anal. 61, No. 1, 161--177 (2023; Zbl 1517.55001)]. Broadly speaking, these algorithms govern the continuous motion of a system under variable external conditions, modeled through a fibration \(p:E\rightarrow B\), where \(B\) represents the external conditions, and each fiber corresponds to the system's constrained configuration space.\N\NThe authors establish both upper and lower bounds for the sequential parametrized topological complexity, employing techniques such as Euler and Stiefel-Whitney characteristic classes. A sharp upper bound is derived in the spirit of \textit{M. Farber} and \textit{M. Grant} in [Proc. Am. Math. Soc. 137, No. 5, 1841--1847 (2009; Zbl 1172.55006)] and subsequently applied to sphere bundles. For vector bundles admitting a complex structure, the authors prove an equality between complexity and a dimensional parameter, raising intriguing questions about the relationship between real and complex structures in vector bundles.\N\NThe cohomology algebra of sphere bundles is analyzed, particularly in cases where the bundle admits a continuous section, building on results from \textit{W. S. Massey} [J. Math. Mech. 7, 265--289 (1958; Zbl 0089.39204)]. Detailed results are presented regarding the cup-length of kernels of diagonal maps, with key findings leading to explicit computations of bounds. The study concludes with examples and applications that showcase the practical implications of the theoretical results, emphasizing cases where upper and lower bounds align, providing complete answers for specific sphere bundles.