Invariance of the Godbillon-Vey class by \(C^1\)-diffeomorphisms for higher codimensional foliations. (English) Zbl 0931.57021

The author shows that the Godbillon-Vey invariant of foliations is invariant by \(C^1\)-diffeomorphisms which preserve the foliations. This is a generalization of former results of G. Raby [Ann. Inst. Fourier 38, No. 1, 205-213 (1988; Zbl 0596.57018)] for codimension-one foliations.
The result seems mysterious when compared with the fact that the Godbillon-Vey class can not be defined for foliations of class \(C^1\) [T. Tsuboi, Ann. Math., II. Ser. 130, No. 2, 227-271 (1989; Zbl 0701.57012)]. In the case of codimension one, we can understand the invariance from the fact that \(C^1\)-conjugacy means that the foliations under consideration are almost \(C^2\)-conjugate [E. Ghys and T. Tsuboi, Ann. Inst. Fourier 38, No. 1, 215-244 (1988; Zbl 0633.58018)]. But we do not have any good explanation for the higher codimensional cases.


57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
57R30 Foliations in differential topology; geometric theory
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