Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Abstract

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.

Inequilogical spaces, directed homology and noncommutative geometry

“…4.8], which gives a strong information on ϑ. It follows that the cubical sets C ϑ have the same classification up to isomorphism [G4,Thm. 4.9] as the C * -algebras A ϑ up to strong Morita equivalence: ϑ is determined up to the action of the linear group GL(2, Z) (I.4.4.1).…”

“…The equivalence of (b) and (e) is a combined result of Pimsner -Voiculescu [PV] and Rieffel [Ri]; that of (c) -(f) has been proved in [G4,Thm. 4.9].…”

“…(C) Applying [G4,Thm. 3.3] to the free action of Z n on the acyclic cubical set A, the algebraic homology of the orbit cubical set is determined as in (2).…”

“…3.3] to the free action of Z n on the acyclic cubical set A, the algebraic homology of the orbit cubical set is determined as in (2). But we want to show that this isomorphism is induced by the inclusion (1), which requires a finer analysis of the arguments on free actions of groups on cubical sets, developed in [G4,Section 3].…”

“…Plainly, ϕ restricts to an injection ϕ: Γ' = Γ" as in (4): if ρ ∈ Γ' = Z n ∩ Γ, the path a ρ (t) = tρ is a directed 1-cube of (R n , ≤ Γ ) and a (positive) cycle modulo Z n . We have to prove that ϕ(Γ') covers Γ" (the argument is similar to the one of [G4,Thm. 4.8]).…”