On the Complexity of Parametrized Motion Planning Algorithms

Abstract.

We study a probabilistic variant of the [Formula: see text]th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On the one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell–Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly different on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups. We also explore several other features of our invariant and its relationships with various other invariants motivated by topological robotics.