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.

%B Journal of Noncommutative Geometry %V 9 %P 1077–1111 %G eng %R 10.4171/JNCG/216 %0 Book Section %B Noncommutative Analysis, Operator Theory and Applications %D 2016 %T Pimsner Algebras and Circle Bundles %A Francesca Arici %A Francesco D'Andrea %A Giovanni Landi %E Alpay, Daniel %E Cipriani, Fabio %E Colombo, Fabrizio %E Guido, Daniele %E Sabadini, Irene %E Sauvageot, Jean-Luc %XWe 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.

%B Noncommutative Analysis, Operator Theory and Applications %I Springer International Publishing %C Cham %P 1–25 %@ 978-3-319-29116-1 %G eng %U https://doi.org/10.1007/978-3-319-29116-1_1 %R 10.1007/978-3-319-29116-1_1 %0 Journal Article %J Journal of Noncommutative Geometry %D 2016 %T Pimsner algebras and Gysin sequences from principal circle actions %A Francesca Arici %A Jens Kaad %A Giovanni Landi %B Journal of Noncommutative Geometry %V 10 %P 29–64 %G eng %U http://hdl.handle.net/2066/162951 %R 10.4171/jncg/228 %0 Thesis %D 2015 %T Principal circle bundles, Pimsner algebras and Gysin sequences %A Francesca Arici %X 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. %I SISSA %G en %1 34744 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Francesca Arici (farici@sissa.it) on 2015-09-26T07:48:48Z No. of bitstreams: 1 AriciThesis.pdf: 983966 bytes, checksum: 4eadf33259d623493f52eaba5c45ec90 (MD5)