Rational sequential parametrized topological complexity
Yuki Minowa
TL;DR
The paper develops a rational-homotopy framework for sequential parametrized topological complexity TC_r[X to B], providing computable algebraic criteria and sharp bounds. It shows that under formality and TNCZ hypotheses, TC_r[X to B] equals the zero-divisors cup-length zcl_r[X to B], and it derives a key odd-degree-extension upper bound TC_r[X to B] ≤ TC_r[\hat X to B] + m(r-1). For fibers elliptic and concentrated in odd degrees, TC_r[X to B] is explicitly (r-1) dim(pi_odd(F)). It also introduces a generating-function analogue F_p(z) and establishes two classes where it is rational with denominator (1−z)^2, linking the growth of TC_r to cat(F) and to the fiber’s rational cohomology structure. These results connect rational homotopy theory with motion-planning under constraints, offering practical tools for computing parametrized TC in rational settings and illuminating the behavior of its growth across r.}
Abstract
Sequential parametrized topological complexity is a numerical homotopy invariant of a fibration, which arose in the robot motion planning problem with external constraints. In this paper, we study sequential parametrized topological complexity in view of rational homotopy theory. We generalize results on topological complexity, and in particular, give an explicit algebraic upper bound for sequential parametrized topological complexity when a fibration admits a certain decomposition, which is a generalization of the result of Hamoun, Rami and Vandembroucq on topological complexity.