The following preprint is available: M. Grandis Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology), Dip. Mat. Univ. Genova, Preprint 480 (May 2003), 29 p. Abstract. This is 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 cohomology agrees with Connes' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of 'noncommutative topology', without the metric information of C*-algebras. MSC: 55U10, 55Nxx, 81R60. Keywords: Cubical sets, combinatorial homology, noncommutative spaces, directed algebraic topology. Available in .pdf and .ps at: http://www.dima.unige.it/~grandis/Bsy.pdf http://www.dima.unige.it/~grandis/Bsy.ps _____ 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/