2021/05/27 by E. Macías-Virgós, Macías-Virgós, E., D. Mosquera-Lois +3
Computer Science · Mathematics · #53C22 #55M30 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Robotic Path Planning Algorithms
paper · pdf · doi:10.48550/arxiv.2105.13006
openalex publication_date 2021/05/27 · openalex created_date 2022/10/19 · openalex updated_date 2026/07/28
We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifol of any Morse-Bott function. This generalizes the Lusternik-Schnirelmann theorem (for Morse functions), and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.