We define quantum lens spaces as ‘direct sums of line bundles’ and exhibit them as ‘total spaces’ of certain principal bundles over quantum projective spaces. For each of these quantum lens spaces we construct an analogue of the classical Gysin sequence in K-theory. We use the sequence to compute the K-theory of the quantum lens spaces, in particular to give explicit geometric representatives of their K-theory classes. These representatives are interpreted as ‘line bundles’ over quantum lens spaces and generically define ‘torsion classes’. We work out explicit examples of these classes.

VL - 9 ER - TY - CHAP T1 - Pimsner Algebras and Circle Bundles T2 - Noncommutative Analysis, Operator Theory and Applications Y1 - 2016 A1 - Francesca Arici A1 - Francesco D'Andrea A1 - Giovanni Landi ED - Alpay, Daniel ED - Cipriani, Fabio ED - Colombo, Fabrizio ED - Guido, Daniele ED - Sabadini, Irene ED - Sauvageot, Jean-Luc AB -We report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate several results with examples of quantum weighted projective and lens spaces and θ-deformations.

JF - Noncommutative Analysis, Operator Theory and Applications PB - Springer International Publishing CY - Cham SN - 978-3-319-29116-1 UR - https://doi.org/10.1007/978-3-319-29116-1_1 ER - TY - JOUR T1 - Pimsner algebras and Gysin sequences from principal circle actions JF - Journal of Noncommutative Geometry Y1 - 2016 A1 - Francesca Arici A1 - Jens Kaad A1 - Giovanni Landi VL - 10 UR - http://hdl.handle.net/2066/162951 ER - TY - THES T1 - Principal circle bundles, Pimsner algebras and Gysin sequences Y1 - 2015 A1 - Francesca Arici AB - 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. PB - SISSA U1 - 34744 U2 - Mathematics U4 - 1 U5 - MAT/07 ER -