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Consider the following two foliations of torus:

1)The Kronecker foliation with slope $\sqrt{2}$

2)The Kronecker foliation with slope $\pi$

As I learn from the literature, these two foliations are not topological equivalent. The proof is that the K theory of their corresponding $C^{*}$ algebras are not isomorphic.In fact two Kronecker foliations with slopes $\alpha$ and $\beta$ are not equivalent if $\alpha$ and $\beta$ are not on the same orbit of action of $Sl_{2}(\mathbb{Z})$. See Non Commutative Geometry by Alain Connes.

But intuitively it is difficult to imagine that these two foliations are different. because in both foliations all leaves are dense! one can not distinguish these two foliations via visible topological behavior.

Is there an intuitive and geometric proof for this fact(without using K theory, $C^{*}$ algebras of foliation,etc.)?

I think, if there is no an intuitive proof, this shows the deep power of the role of $C^{*}$ algebra of foliations, at least in this example.

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Just what the doctor ordered, a proof using diffeology instead of $\mathrm C^*$-algebras:

MR0799609 Donato, Paul(F-CNRS-T); Iglésias, Patrick(F-CNRS-T) Exemples de groupes difféologiques: flots irrationnels sur le tore. (French. English summary) [Examples of diffeological groups: irrational flows on the torus] C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 4, 127–130.

Author summary (translated from the French): "We illustrate J.-M. Souriau's technique of `diffeological spaces and groups' in the case of the quotient $T_α$ of the standard torus by the irrational flow with slope $α$. Computing the universal covering $\mathbf R$ and the fundamental group $\mathbf Z^2$ of $T_α$ allows us to classify these tori diffeologically: $T_α$ and $T_β$ are diffeomorphic if and only if $α∼β$ modulo $\mathrm{GL}(2,\mathbf Z)$; moreover, the computation of $\mathrm{Diff}(T_α)$ reveals a difference between quadratic irrationals and other irrationals.''

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  • $\begingroup$ Thank you again for the interesting paper. I confess that I did not read it in details. But I review it. I think they show the foliations are not smooth equivalent, they do not show they are not topological equivalent.Am I right? $\endgroup$ – Ali Taghavi Sep 30 '14 at 17:47

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