Codimension one foliations with Bott-Morse singularities I

Abstract

We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among other things, a good topological understanding of the leaves one may have around a center-type component of the singular set, and also of the topology of its basin. The stability theorem further allows the description of the topology of the boundary of the basin and how the topology of the leaves changes when passing from inside to outside the basin. This is described via fiberwise Milnor-Wallace surgery. A keypoint for this is to show that if the boundary of the basin of a center is non-empty, then it contains a saddle; in this case we say that the center and the saddle are paired. We then describe the possible pairings one may have in dimension three and use a construction motivated by the classical saddle-node bifurcation, that we call foliated surgery, that allows the reduction of certain pairings of singularities of a foliation. This is used together with our previous work on the topic to prove an extension for 3-manifolds of Reeb's sphere recognition theorem. ContentsSubject Classification: Primary 57R30, 58E05; Secondary 57R70, 57R45.

Codimension 1 foliations with Bott-Morse singularities II

“…Now, we go to the boundary ∂C(N 0 , F) of this set. By construction, if ∂C(N 0 , F) = ∅, then M = C(N 0 , F), there are no saddle singularities of F in M and we are in the situation previously envisaged in [19]. Otherwise, there must be a saddle component N 1 in ∂C(N 0 , F) and no center components there; in this case, we say that N 0 and N 1 are paired, a key-concept for this article, inspired by Camacho and Scárdua [3].…”

“…The concept of foliations with Bott-Morse singularities on smooth manifolds obviously comes from Reeb [17], from the landmark work of Morse in his Colloquium Publication [16], as well as from Bott's generalization of Morse functions in [1]. This notion for foliations was introduced in our previous article [19], where we focused on the case in which all singularities were transversally of center-type. The presence of saddles obviously makes the theory richer, and this is the situation we now envisage.…”

On singular foliations on the solid torus

“…Singular foliations can be defined in different ways and have been studied by several authors (see [6,17,18]). For a recent account of the theory see for example [1,3,11,12,15,16]. We use as definition the one given in [14]: a C r -foliation F , r 1, of an m-manifold M is a partition of M in connected immersed C r submanifolds, called leaves, such that the module X r (F ) of the C r vector fields of M tangent to the leaves is transitive, that is, given p ∈ M and v ∈ T p L, where L is the leaf by p, there exists X ∈ X r (F ) such that X(p) = v. This definition is equivalent to those stated in [17] and [18].…”

“…Camacho and Scárdua considered in [3] codimension one foliations with isolated singularities of Morse type. Scárdua and Seade studied in [15] and [16] codimension one transversally oriented foliation on oriented closed manifolds having non-empty compact singular set which is locally defined by Bott-Morse functions. We restrict our investigation to a seemingly basic situation, namely C 2 -foliations on the solid torus S 1 × D 2 which have L 0 = S 1 × {0} as their only singular leaf, and L 1 = S 1 × ∂D 2 as their only compact regular leaf.…”

“…The family of such foliations will be denoted by A. To quote L. Conlon from the MathSciNet review of [16]: "Foliations with singularities are a real zoo, but they do arise in nature (e.g., the orbit foliation of a Lie group action). In order to get any reasonable structure theory, it is necessary to severely restrict the types of singularities."…”