The following preprint is available in ps: Normed combinatorial homology and noncommutative tori Marco Grandis Dip. Mat. Univ. Genova, Preprint 484 (Jul 2003), 14 p. http://www.dima.unige.it/~grandis/Bsy2.ps Abstract. Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of "noncommutative topology", agreeing with some results of noncommutative geometry but lacking the metric aspects of C*-algebras. Here, we make such similarity 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. MSC: 55U10, 81R60, 55Nxx. Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups. Marco Grandis Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/