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We study codimension 1 foliations with singularities defined locally by Bott‐Morse functions on closed oriented manifolds. We extend 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 about a center‐type component of the singular set, and also of the topology of its  basin . The stability theorem further permits one to determine 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 in this respect 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, which we call  foliated surgery , that allows for 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.

Codimension 1 foliations with Bott–Morse singularities II