Bridge Cohomology: a Generalization of Hochschild and Cyclic Cohomologies

The connection between Hochschild and cyclic cohomologies with generalized De Rham homology and index theories for arbitrary algebras has long been established by the work of Connes, Karoubi, Loday, Feigin, Tsygan, et al. Here we generalize these cohomology theories even further, essentially creating a theory that establishes a step-wise bridge between the two. Motivation for this construction comes from trying to generalize the Hochschild-Kostant-Rosenberg-Connes theorem to manifolds with boundary, and applications in tracial constructions in certain classes of pseudodifferential operators. The situation that arises is a subcomplex of Hochschild functionals that are not cyclic, but rather descend to a subcomplex under the cyclic operator. The complexes that arise from this are developed at length, ending with Gysin-Connes sequences that relate bridge cohomology to Hochscild and cyclic cohomologies, as well as the theorem of excision. Further geometric and topological interests of this theory include extending Chern-Weil theory to manifolds with boundary via pairings between bridge cohomology and higher K-theories.