Directed combinatorial homology and noncommutative geometry

We will present a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologically-trivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed homology agrees with Connes' analysis in noncommutative geometry [C1]. Thus, cubical sets can provide a sort of 'noncommutative topology', without the metric information of C*-algebras [G1]. This similarity can be made stricter by introducing normed cubical sets and their normed directed homology, formed of normed preordered abelian groups. The normed cubical sets associated with irrational rotations have thus the same classification up to isomorphism as the well-known irrational rotation C*-algebras [G2]. Finally, we will see that part of these results can also be obtained with a different approach, based on D. Scott's equilogical spaces [Sc] and developed in [G3, G4]. Comments printed in gray characters can be omitted. The index α takes values 0, 1, also written as –, + (e.g. in superscripts).