On the Hurder-Katok extension of the Godbillon-Vey invariant. (English) Zbl 0722.57012

Tsuboi, Takashi

J. Fac. Sci., Univ. Tokyo, Sect. I A 37, No. 2, 255-262 (1990).

S. Hurder and A. Katok [Publ. Math., Inst. Hautes Étud. Sci. 72, 5-61 (1990)] defined the Godbillon-Vey invariant for foliations of class \(C^{1+\alpha}\), \(\alpha >\). The author considers the case \(0<\alpha <\) and proves the following two theorems: 1. The Godbillon-Vey 2-cocycle defined in \(Diff_+^{\omega}(S^ 1)\) is not continuous in the \(C^{1+\alpha}\) topology for \(0<\alpha <\). 2. For \(0<\alpha <\), there is a foliated R-product F with compact support over a closed oriented surface \(\Sigma\) of class \(C^{1+\alpha}\) with the following properties. F admits a partition into a countable number of saturated Borel sets \(B_ i\) where Godbillon-Vey invariants \(GV(F,B_ i)\) are defined and \(\Sigma GV(F,B_ i)=\infty\).

Reviewer: W.Mozgawa (Lublin)

MSC:

57R30	Foliations in differential topology; geometric theory
53C12	Foliations (differential geometric aspects)

Keywords:

Godbillon-Vey classes; foliations; foliated R-product; closed oriented surface; Borel sets