Hopf cyclic cohomology and transverse characteristic classes

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand-Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of foliation groupoids.

💡 Research Summary

The paper develops a refined cyclic cohomological framework for computing the Hopf‑cyclic cohomology of Hopf algebras that arise from infinite primitive Cartan‑Lie pseudogroups, and for transferring their characteristic classes to foliations. The authors begin by recalling the classical Gelfand‑Fuks complex, which encodes the cohomology of the Lie algebra of formal vector fields, and the Bott complex for equivariant cohomology, which provides a geometric model for transverse characteristic classes. A canonical chain map φ from the Gelfand‑Fuks complex to the Bott complex is constructed; this map has been known abstractly but its concrete image has not been identified within a Hopf‑cyclic setting.

The central achievement of the work is the precise identification of the image of φ as a Hopf‑cyclic complex C(H) associated with the geometric Hopf algebra H generated by the pseudogroup. By showing that φ respects the Hopf‑module structure and intertwines the Hochschild differential d with the Connes operator B, the authors prove an isomorphism of cochain complexes \