Characteristic classes of foliations via SAYD-twisted cocycles
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We find the first non trivial “SAYD-twisted” cyclic cocycle over the groupoid action algebra under the symmetry of the affine 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.