Abstract
AbstractWe 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 submanifolds 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.
Funder
MINECO Spain
Consellería de Economía, Emprego e Industria, Xunta de Galicia
Ministerio de Ciencia, Innovación y Universidades
Xunta de Galicia
Publisher
Springer Science and Business Media LLC
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