On smooth deformations of foliations with singularities

Journal of Singularities
volume 9 (2014), 101-110

Received 30 May 2012. Received in revised form 18 March 2014.

DOI: 10.5427/jsing.2014.9i

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Abstract:

We study smooth deformations of codimension one foliations with Morse and Bott-Morse singularities of center-type. We show that in dimensions at least 3, every small smooth deformation by foliations of a Morse function with only center type singularities is a deformation by Morse functions. We also show that this statement is false in dimension 2. In the same vein we show that if F is a foliation with Bott-Morse singularities on a manifold M, all of center type, and if we assume there is a component N of the singular locus of f of codimension at least 3 such that H^1(N,R)=0, then every small smooth deformation F_t of F is compact, stable and given by a Bott-Morse function f_t: M -> [0,1] with only two critical values at 0 and 1. Furthermore, each such foliation F_t is topologically equivalent to F. Hence, Bott-Morse foliations with only center-type singularities and having a component N of the singular locus of F of codimension m at least 3 such that H^1(N,R)=0$, are structurally stable under smooth deformations. These statements are false in general if we drop the codimension m at least 3 condition.