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In [C2], Baum-Connes state a conjecture for the K-theory of C*-algebras of foliations. This conjecture has been proved by T. Natsume [N2] for C°°-codimension one foliations without holonomy on a closed manifold. We propose here another proof of the conjecture for this class of foliations, more geometric and based on the existence of the Thom isomophism, proved by A. Connes in [C3]. The advantage of this approach is that the result will be valid for all C'-foliations .

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Abstract. We demonstrate that any minimal transversely Cantor compact lamination of dimension p and class C 1 without holonomy is an inverse limit of compact branched manifolds of dimension p. To prove this result, we extend the triangulation theorem for C 1 manifolds to transversely Cantor C 1 laminations. In fact, we give a simple proof of this classical theorem based on the existence of C 1 -compatible differentiable structures of class C ∞ .