Codimension one foliations with Bott-Morse singularities II

TL;DR

This paper extends the theory of codimension one foliations with Bott-Morse singularities on closed manifolds, introducing stability results, topological descriptions of leaves and basins, and a classification of singularity pairings, including a new foliated surgery technique.

Contribution

It develops a stability theorem for Bott-Morse singularities, describes the topology of leaves and basins, and introduces foliated surgery for singularity reduction, extending Reeb's sphere recognition to 3-manifolds.

Findings

Stability theorem for Bott-Morse foliations

Topological characterization of leaves around center-type singularities

Extension of Reeb's sphere recognition theorem to 3-manifolds

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 key-point 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.