K-theory for the leaf space of foliations by Reeb components

The K-theory of the C¹-algebra $\text{C}{}^1\text{(V, F)}$ associated to C^∞-foliations (V, F) of a manifold V in the simplest non-trivial case, i.e., dim V = 2, is studied. Since the case of the Kronecker foliation was settled by Pimsner and Voiculescu (J. Operator Theory4 (1980), 93–118), the remaining problem deals with foliations by Reeb components. The K-theory of $\text{C}{}^1\text{(V, F)}$ for the Reeb foliation of S³ is also computed. In these cases the C¹-algebra $\text{C}{}^1\text{(V, F)}$ is obtained from simpler C¹-algebras by means of pullback diagrams and short exact sequences. The K-groups $\text{K}{}_1\text{(C}{}^1\text{(V, F))}$ are computed using the associated Mayer-Vietoris and six-term exact sequences. The results characterize the C¹-algebra of the Reeb foliation of $\text{T}$² uniquely as an extension of C(S¹) by C(S¹). For the foliations of $\text{T}$² it is found that the K-groups count the number of Reeb components separated by stable compact leaves. A C^∞-foliation of $\text{T}$² such that K₁(C¹($\text{T}$², F)) has infinite rank is also constructed. Finally it is proved, by explicit calculation using (M. Penington, “K-Theory and C¹-Algebras of Lie Groups and Foliations,” D. Phil. thesis, Oxford, 1983), that the natural map $\text{μ: K}{}_1\text{,}{}_{\mathrm{\tau }}\text{(BG) → K}{}_1\text{(C}{}^1\text{(V, F))}$ is an isomorphism for foliations by Reeb components of $\text{T}$² and S³. In particular this proves the Baum-Connes conjecture (P. Baum and A. Connes, Geometric K-theory for Lie groups, preprint, 1982; A. Connes, Proc. Symp. Pure Math.38 (1982), 521–628) when V = $\text{T}$².