Fiber Bundles and Foliations

Abstract

The notion of holonomy of a leaf introduced in the previous chapter is essentially of local character. It is defined by a group of germs of diffeomorphisms of a transverse section to a leaf, with a fixed point. In certain circumstances, however, it is possible to associate to the foliation a group of diffeomorphisms of a global transverse section, containing in a certain well-defined sense the holonomy of each leaf. This is the case of foliations whose leaves meet transversely all the fibers of a fiber bundle E. The importance of these foliations is in the fact that they are characterized by their holonomy, in this case given by a representation ϕ : π ₁ (B) ➞ Diff (F) of the fundamental group of the base of E to the group of diffeomorphisms of the fiber of E. In this manner properties of ϕ translate to properties of the foliation. For example, the action ϕ has exceptional minimal sets if and only if the same occurs for the foliation. Sacksteder’s example, of a C ^∞ codimension one foliation with an exceptional minimal set, is a typical case of what we will see in this chapter.