# Journal of Noncommutative Geometry

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**Volume 10, Issue 1, 2016, pp. 29–64**

**DOI: 10.4171/JNCG/228**

Published online: 2016-03-22

Pimsner algebras and Gysin sequences from principal circle actions

Francesca Arici^{[1]}, Jens Kaad

^{[2]}and Giovanni Landi

^{[3]}(1) Radboud University Nijmegen, Netherlands

(2) Radboud University Nijmegen, Netherlands

(3) Università di Trieste, Italy

A self Morita equivalence over an algebra $B$, given by a $B$-bimodule $E$, is thought of as a line bundle over $B$. The corresponding Pimsner algebra $\mathcal O_E$ is then the total space algebra of a noncommutative principal circle bundle over $B$. A natural Gysin-like sequence relates the $KK$-theories of $\mathcal O_E$ and of $B$. Interesting examples come from $\mathcal O_E$ a quantum lens space over $B$ a quantum weighted projective line (with arbitrary weights). The $KK$-theory of these spaces is explicitly computed and natural generators are exhibited.

*Keywords: *KK-theory, Pimsner algebras, Gysin sequences, circle actions, quantum principal bundles, quantum lens spaces, quantum weighted projective spaces

Arici Francesca, Kaad Jens, Landi Giovanni: Pimsner algebras and Gysin sequences from principal circle actions. *J. Noncommut. Geom.* 10 (2016), 29-64. doi: 10.4171/JNCG/228