3 - Two classical theorems

Summary

In this chapter we present two milestones of the early theory of foliations, namely the theorems of Haefliger and of Novikov. Both theorems concern foliations of codimension 1 on three-dimensional manifolds.

Haefliger's theorem dates from the late 1950s, and concerns the problem of constructing codimension 1 foliations on 3-manifolds. One version asserts that if a compact three-dimensional manifold carries an analytic foliation of codimension 1, then this manifold must have infinite fundamental group. Thus, such a foliation cannot exist on the 3-sphere, for example. We will present a detailed proof, which is close to Haefliger's original argument, and which involves various notions of independent interest. The first of these is that of a Morse function into a manifold carrying a codimension 1 foliation, which we discuss in Subsection 3.1.2, after having reviewed the classical theory of Morse functions into the line. The other is that of foliations with isolated singularities on a twodimensional disk, to be discussed in Subsection 3.1.3. These singular foliations arise by pull-back along a Morse function from the disk into a given 3-manifold equipped with a codimension 1 foliation.

Novikov's theorem dates from the 1960s, and concerns the existence of compact leaves. Explicitly, it states that any (smooth) transversely orientable codimension 1 foliation of a compact manifold with finite fundamental group must have a compact leaf. Moreover, a closer analysis reveals that this compact leaf must be a torus, and that inside this torus, the given foliation looks exactly like the Reeb foliation discussed in Example 1.1 (5). We will also present a detailed proof of this result of Novikov's.