Homotopic distance and generalized motion planning

TL;DR

This paper establishes bounds on homotopic distance using Morse-Bott functions and applies these results to generalized motion planning, extending classical theorems and providing new tools for navigation problems.

Contribution

It generalizes the Lusternik-Schnirelmann theorem and topological complexity bounds to broader settings involving Morse-Bott functions and cut loci.

Findings

Bound on homotopic distance using Morse-Bott functions

Extension of classical theorems to analytic manifolds

Application of navigation functions to generalized motion planning

Abstract

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.