Proper topological complexity

TL;DR

This paper introduces the proper topological complexity, a new homotopy invariant for configuration spaces that accounts for avoiding unsafe regions during motion planning.

Contribution

It defines and characterizes the proper topological complexity as an exterior sectional category, extending classical invariants to safer motion planning scenarios.

Findings

Proper topological complexity is homotopy functorial.

It is characterized as an exterior sectional category.

The invariant is deeply analyzed within exterior homotopy theory.

Abstract

We introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe'' area. To make it a homotopy functorial invariant we characterize it as a particular instance of the exterior sectional category of an exterior map, an invariant of the exterior homotopy category which is also deeply analyzed.