Homotopic distance between maps

TL;DR

This paper introduces the homotopic distance between maps, unifying concepts like Lusternik-Schnirelmann category and topological complexity, leading to new insights and simplified proofs of their properties.

Contribution

It defines homotopic distance as a general invariant, encompassing existing invariants, and demonstrates its utility in deriving properties and new results.

Findings

Homotopic distance generalizes Lusternik-Schnirelmann category and topological complexity.

Unified proofs of properties for these invariants.

New results derived from the homotopic distance framework.

Abstract

We show that both Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.