Principal circle bundles, Pimsner algebras and Gysin sequences

Principal circle bundles and Gysin sequences play a crucial role in mathematical physics, in particular in Chern-Simons theories and T-duality. This works focuses on the noncommutative topology of principal circle bundles: we investigate the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. At the C*-algebraic level, we start from a self-Morita equivalence bimodule E for a C*-algebra B which we think of as a non commutative line bundle over the `base space’ algebra B. The corresponding Pimsner algebra O_E, is then the total space algebra of an associated circle bundle. A natural six term exact sequence, an analogue of the Gysin sequence for circle bundles, relates the KK-theories of O_E and of the base space B. We illustrate several results with the examples of quantum weighted projective and lens spaces.