The Mobius Band and the Mobius Foliation

Some years ago we introduced a new topological invariant for foliated manifolds using techniques from noncommutative geometry, in particular the pairing between K-Theory and cyclic cohomology. The motivation came from flat principal G-bundles where the base space is a non simply connected manifold. The computation of this invariant is quite complicated. In this article we try to perform certain computations for the Mobius band (or Mobius foliation) which is an interesting nontrivial example of foliations; this example has a key feature: it is the simplest case of a large class of examples of foliations, that of bundles with discrete structure groups which also includes the foliations given by flat vector (or G-principal) bundles. We shall see that the Mobius foliation example also helps one to understand another large class of examples of foliations coming from group actions on manifolds which are not free. PACS classification: 11.10.-z; 11.15.-q; 11.30.-Ly