Alcides Lins Neto

The topology of integrable differential forms near a singularity

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The Topology of the Leaves

We saw in the previous chapter that the leaves of a C r foliation inherit a C r differentiate manifold structure immersed in the ambient manifold. In this chapter we will study the topological properties of these immersions, giving special emphasis to the asymptotic properties of the leaves.

Fiber Bundles and Foliations

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.

Novikov’s Theorem

The following theorem, due to Novikov [40], is one of the deepest, most beautiful theorems in foliations.

Topological Aspects of the Theory of Group Actions

In this chapter M denotes a C ∞ differentiable manifold and G a simply connected Lie group.

Holonomy and the Stability Theorems

In this chapter F denotes a foliation of codimension n and class C r , r ≥ 1, of a manifold M m . Our objective is to study the behavior of the leaves near a fixed compact leaf F. By the transverse uniformity of F it is sufficient to study the first returns of leaves to a small transverse section Σ of dimension n passing through a point p ∈ F. For each closed path γ in F passing through p, these returns can be expressed by a local C r diffeomorphism of Σ, f γ, with f γ (p) = p and where for x ∈ Σ sufficiently near p, f γ(x) is the first return “over γ” of the leaf of F which passes through x.

Analytic Foliations of Codimension One

A codimension n foliation F of an m-dimensional manifold is analytic when the change of coordinate maps which define F are analytic local diffeomorphisms of ℝ m . Under these conditions any element of the holonomy of a leaf of F has a representation which is an analytic local diffeomorphism of ℝ n .