On directed homotopy equivalences and a notion of directed topological complexity

TL;DR

This paper introduces a new concept of directed homotopy equivalence and directed topological complexity, establishing their properties and invariance, and comparing them to existing notions in directed topology.

Contribution

It proposes a novel framework for directed homotopy equivalence and directed topological complexity with desirable invariance properties and compares it to previous approaches.

Findings

Directed homotopy equivalence implies bisimilar natural homologies.

Directed topological complexity is invariant under the new equivalence.

Being dicontractible is characterized by directed topological complexity one.

Abstract

This short note introduces a notion of directed homotopy equivalence and of "directed" topological complexity (which elaborates on the notion that can be found in e.g. Farber's book) which have a number of desirable joint properties. In particular, being dihomotopically equivalent implies having bisimilar natural homologies (defined in Dubut et al. 2015). Also, under mild conditions, directed topological complexity is an invariant of our directed homotopy equivalence and having a directed topological complexity equal to one is (under these conditions) equivalent to being dihomotopy equivalent to a point (i.e., to being "dicontractible", as in the undirected case). It still remains to compare this notion with the notion introduced in Dubut et al. 2016, which has lots of good properties as well. For now, it seems that for reasonable spaces, this new proposal of directed homotopy equivalence identifies more spaces than the one of Dubut et al. 2016.