Homotopic Distance and Generalized Motion Planning
Explore the concept of homotopic distance between two maps in this 41-minute lecture from the Applied Algebraic Topology Network. Delve into how Lusternik-Schnirelmann category and topological complexity are specific instances of this broader notion, allowing for unified proofs of various properties and the emergence of new results. Learn about the theorem that bounds the homotopic distance between two maps on a manifold by the sum of their relative distances on critical submanifolds of any Morse-Bott function. Discover how this generalizes the Lusternik-Schnirelmann theorem for Morse functions and Farber's result for topological complexity. Explore the practical application of these concepts in solving generalized motion planning problems using navigation functions.