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Gysin Exact Sequences for Quantum Weighted Lens Spaces

Part of the MATRIX Book Series book series (MXBS,volume 1)

Abstract

We describe quantum weighted lens spaces as total spaces of quantum principal circle bundles, using a Cuntz-Pimsner model. The corresponding Pimsner exact sequence is interpreted as a noncommutative analogue of the Gysin exact sequence. We use the sequence to compute the K-theory and K-homology groups of quantum weighted lens spaces, extending previous results and computations due to the author and collaborators.

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References

  1. Arici, F., Rennie, A.: Explicit isomorphism of mapping cone and Cuntz-Pimsner exact sequences (2016). arXiv:1605.08593

    Google Scholar 

  2. Arici, F., Brain, S., Landi, G.: The Gysin sequence for quantum lens spaces. J. Noncommut. Geom. 9, 1077–1111 (2015)

    MathSciNet  CrossRef  Google Scholar 

  3. Arici, F., D’Andrea, F., Landi, G.: Pimsner algebras and circle bundles. In: Noncommutative Analysis, Operator Theory and Applications, vol. 252, pp. 1–25. Birkhäuser, Cham (2016)

    MATH  Google Scholar 

  4. Arici, F., Kaad, J., Landi, G.: Pimsner algebras and Gysin sequences from principal circle actions. J. Noncommut. Geom. 10, 29–64 (2016)

    MathSciNet  CrossRef  Google Scholar 

  5. Blackadar, B.: K-Theory for Operator Algebras, 2nd edn. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  6. Brzeziński, T., Fairfax, S.A.: Quantum teardrops. Commun. Math. Phys. 316, 151–170 (2012)

    MathSciNet  CrossRef  Google Scholar 

  7. Brzeziński, T., Fairfax, S.A.: Notes on quantum weighted projective spaces and multidimensional teardrops. J. Geom. Phys. 93, 1–10 (2015)

    MathSciNet  CrossRef  Google Scholar 

  8. Brzeziński, T., Szymański, W.: The C*-algebras of quantum lens and weighted projective spaces (2016). arXiv:1603.04678

    Google Scholar 

  9. Cuntz, J.: Simple C- algebras generated by isometries. Commun. Math. Phys. 57, 173–185 (1977)

    MathSciNet  CrossRef  Google Scholar 

  10. Cuntz, J., Krieger, W.: A class of C-algebras and topological Markov chains. Invent. Math. 56, 251–268 (1980)

    MathSciNet  CrossRef  Google Scholar 

  11. D’Andrea, F., Landi, G.: Quantum weighted projective and lens spaces. Commun. Math. Phys. 340, 325–353 (2015)

    MathSciNet  CrossRef  Google Scholar 

  12. Eilers, S., Restorff, G., Ruiz, E., Sørensen, A.P.W.: Geometric classification of graph C-algebras over finite graphs (2016). arXiv:1604.05439

    Google Scholar 

  13. Exel, R.: Circle actions on C-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence. J. Funct. Anal. 122, 361–401 (1994)

    MathSciNet  CrossRef  Google Scholar 

  14. Gabriel, O., Grensing, M.: Spectral triples and generalized crossed products (2013). arXiv:1310.5993

    Google Scholar 

  15. Goffeng, M., Mesland, B., Rennie, A.: Shift-tail equivalence and an unbounded representative of the Cuntz-Pimsner extension. Ergod. Theory Dyn. Syst. (2015, to appear). arXiv:1512.03455

    Google Scholar 

  16. Hong, J.H., Szymański, W.: Quantum lens spaces and graph algebras. Pac. J. Math. 211, 249–263 (2003)

    MathSciNet  CrossRef  Google Scholar 

  17. Kajiwara, T., Pinzari, C., Watatani, Y.: Ideal structure and simplicity of the C–algebras generated by Hilbert bimodules. J. Funct. Anal. 159, 295–322 (1998)

    MathSciNet  CrossRef  Google Scholar 

  18. Karoubi, M.: K-Theory: An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 226. Springer, Berlin (1978)

    Google Scholar 

  19. Montgomery, S.: Hopf Algebras and Their Actions on Rings. Regional Conference Series in Mathematics, vol. 82. American Mathematical Society, Providence, RI (1993)

    Google Scholar 

  20. Pimsner, M.: A class of C -algebras generalising both Cuntz-Krieger algebras and crossed products by \(\mathbb {Z}\). In: Free Probability Theory. Fields Institute Communications, vol. 12, pp. 189–212. American Mathematical Society, Providence, RI (1997)

    Google Scholar 

  21. Rennie, A., Robertson, D., Sims, A.: The extension class and KMS states for Cuntz-Pimsner algebras of some bi-Hilbertian bimodules. J. Topol. Anal. 09(02), 297 (2015). https://doi.org/10.1142/S1793525317500108

    MathSciNet  CrossRef  Google Scholar 

  22. Sitarz, A., Venselaar, J.J.: The Geometry of quantum lens spaces: real spectral triples and bundle structure. Math. Phys. Anal. Geom. 18, 1–19 (2015)

    MathSciNet  CrossRef  Google Scholar 

  23. Vaksman, L., Soibelman, Ya.: The algebra of functions on the quantum group SU(n + 1) and odd-dimensional quantum spheres. Leningr. Math. J. 2, 1023–1042 (1991)

    Google Scholar 

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Acknowledgements

We thank the mathematical research institute MATRIX in Australia and the organisers of the workshop “Refining C-algebraic invariants using KK-theory”, where part of this research was performed. This work was motivated by discussions with Efren Ruiz about the structure of graph algebras. We thank Adam Rennie for helpful discussion and for his hospitality at the University of Wollongong, where part of this work was carried out. Finally, the author would like to thank Francesco D’Andrea, Giovanni Landi, Bram Mesland and Walter van Suijlekom for helpful comments on an early version of this work. This research was partially supported by NWO under the VIDI-grant 016.133.326.

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Correspondence to Francesca Arici .

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Arici, F. (2018). Gysin Exact Sequences for Quantum Weighted Lens Spaces. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_12

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