The Godbillon–Vey cyclic cocycle for PL-foliations

Abstract

We define a cyclic cocycle which corresponds to the piecewise linear Godbillon–Vey class of Ghys and Sergiescu [E. Ghys, V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987) 185–239]. Using Connes's pairing [A. Connes, Non-commutative differential geometry. Part II: De Rham homology and noncommutative algebra, Publ. Math. Inst. Hautes Études Sci. 62 (1985) 257–360; A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in: H. Araki, G. Effros (Eds.), Geometric Methods in Operator Algebras, Pitman Res. Notes Math. Ser., vol. 123, Longman, Harlow, 1986, pp. 52–144] between cyclic cohomology and K-theory, we then evaluate this cocycle on a suitable K-theory class and obtain a nontrivial result, for foliations of the 3-torus by slope components.