# Directed topological complexity

## TL;DR

This paper extends the concept of topological complexity to directed topological spaces, addressing motion planning problems with control constraints and applications in concurrency theory.

## Contribution

It adapts topological complexity to directed spaces, explores its properties, and provides calculations for specific classes, linking to directed homotopy theory.

## Key findings

- Directed topological complexity can be defined and computed for certain classes of spaces.
- The adapted notion applies to controlled systems and concurrency models.
- Applications include understanding directed homotopy equivalences.

## Abstract

It has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced by M. Farber. The above notion fits the motion planning problem of robotics when there are no constraints on the actual control that can be applied to the physical apparatus. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential future dynamics. In this paper we adapt the notion of topological complexity to the case of directed topological spaces, which encompass such controlled systems, and also systems which appear in concurrency theory. We study its first properties, make calculations for some interesting classes of spaces, and show applications to a form of directed homotopy equivalence.