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

Abstract

The K-theory of the C ∗-algebra C ∗(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 Theory 4 (1980) , 93–118), the remaining problem deals with foliations by Reeb components. The K-theory of C ∗(V, F) for the Reeb foliation of S 3 is also computed. In these cases the C ∗-algebra C ∗(V, F) is obtained from simpler C ∗-algebras by means of pullback diagrams and short exact sequences. The K-groups K ∗(C ∗(V, F)) are computed using the associated Mayer-Vietoris and six-term exact sequences. The results characterize the C ∗-algebra of the Reeb foliation of T 2 uniquely as an extension of C( S 1) by C( S 1). For the foliations of T 2 it is found that the K-groups count the number of Reeb components separated by stable compact leaves. A C ∞-foliation of T 2 such that K ∗( C ∗( T 2, F)) has infinite rank is also constructed. Finally it is proved, by explicit calculation using ( M. Penington, “ K-Theory and C ∗-Algebras of Lie Groups and Foliations,” D. Phil. thesis, Oxford, 1983 ), that the natural map μ: K ∗, τ(BG) → K ∗(C ∗(V, F)) is an isomorphism for foliations by Reeb components of T 2 and S 3. 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 = T 2.