  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.
