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On the Hurder-Katok extension of the Godbillon-Vey invariant. (English) Zbl 0722.57012
S. Hurder and A. Katok [Publ. Math., Inst. Hautes √Čtud. 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)

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