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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 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)