Chapter III - Tangential Cohomology

Summary

In this chapter we discuss certain cohomology groups associated with a foliated space, which we shall call tangential cohomology groups. It will be in these groups that invariants connected with the index theorem will live. Similar groups have been considered, for instance, in [Kamber and Tondeur 1975; Molino 1973; Vaisman 1973; Sarkaria 1978; Heitsch 1975; El Kacimi-Alaoui 1983; Haefliger 1980] (we discuss Haefliger's work at the end of chapter IV). The similarities and differences between the three situations are easy to describe; all involve differential forms which are smooth in the tangential direction of the foliation. The difference comes in the assumptions on the transverse behavior: for foliated manifolds (Kamber, Tondeur, and others), forms are C^∞ in the transverse direction; for foliated spaces (the present treatment), the forms are to be continuous in the transverse directions, since that is all that makes sense; and finally for foliated measure spaces, the forms are to be measurable in the transverse direction, for again that is all that makes sense.

Thus, we let X be a metrizable foliated space with foliation tangent bundle FX → X, as defined in Chapter II. The quickest and simplest way to introduce the tangential cohomology is via sheaf theory and sheaf cohomology, but for those readers who are not familiar with such notions we show how to define the groups via a de Rham complex and also show in an appendix how to give a completely algebraic definition.