Journal of Noncommutative Geometry

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Volume 9, Issue 3, 2015, pp. 965–998
DOI: 10.4171/JNCG/213

Published online: 2015-10-29

Characteristic classes of foliations via SAYD-twisted cocycles

Bahram Rangipour[1] and Serkan Sütlü[2]

(1) University of New Brunswick, Fredericton, Canada
(2) Işık University, Istanbul, Turkey

We find the first non trivial “SAYD-twisted” cyclic cocycle over the groupoid action algebra under the symmetry of the ane linear transformations of the Euclidian space. We apply the cocycle to construct a characteristic map by which we transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. In codimension 1, our result matches with the (only explicit) computation done by Connes–Moscovici. We carry out the explicit computation in codimension 2 to present the transverse fundamental class, the Godbillon–Vey class, and the other four residual classes as cyclic cocycles on the groupoid action algebra.

Keywords: Connes–Moscovici Hopf algebras, Hopf cyclic cohomology, cyclic cohomology, Weil algebra, characteristic classes of foliations

Rangipour Bahram, Sütlü Serkan: Characteristic classes of foliations via SAYD-twisted cocycles. J. Noncommut. Geom. 9 (2015), 965-998. doi: 10.4171/JNCG/213