On directed homotopy equivalences and a notion of directed topological complexity

This short note introduces a notion of directed homotopy equivalence (or dihomotopy equivalent) and of “directed” topological complexity (which elaborates on the notion that can be found in e.g. [9]) which have a number of desirable joint properties. In particular, being dihomotopically equivalent implies having bisimilar natural homologies (defined in [5]). 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 [8], 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 [8].