Skip to main content
Log in

K-theory of noncommutative Bieberbach manifolds

  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

We compute the K-theory of noncommutative Bieberbach manifolds, which are fixed point C* subalgebras of a three-dimensional noncommutative torus by a free action of a cyclic group ℤ N , N = 2, 3, 4, 6.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. L. Bieberbach, “Über die Bewegungsgruppen der Euklidischen Räume”, Math. Ann. 70 (3), 297–336 (1911).

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Bieberbach, “Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich”, Math. Ann. 72 (3), 400–412, (1912).

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Blackadar, K-Theory for Operator Algebras, MSRI Publ. 5, 2nd ed. (Cambridge University Press, Cambridge, 1998).

    MATH  Google Scholar 

  4. J. Buck and S. Walters, “Connes–Chern Characters Of Hexic And Cubic Modules,” J. Operator Theory 57 (1), 35–65 (2007).

    MATH  MathSciNet  Google Scholar 

  5. J. Buck and S. Walters, “Non commutative spheres associated with the hexic transform and their K-theory, J. Operator Theory 58 (2), 441–462 (2007).

    MATH  MathSciNet  Google Scholar 

  6. S. Echterhoff, W. Lück, N. C. Phillips, and S. Walters, “The structure of crossed products of irrational rotation algebras by finite subgroups of SL 2(Z),” J. Reine Angew. Math. (Crelle’s Journal) 639, 173–221 (2010).

    MATH  Google Scholar 

  7. S. Echterhoff, R. Nest, H. Oyono-Oyono, “Fibrations with noncommutative fibers,” J. Noncommut. Geom. 3 (3), 377–417 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Kishimoto and H. Takai, “Some Remarks on C*-Dynamical Systems with a Compact Abelian Group,” Publ. RIMS, Kyoto Univ. 14 (2), 383–397 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Kumjian, “On the K-theory of the symmetrized non-commutative torus,” C. R. Math. Rep. Acad. Sci. Canada 12 (2–3), 87–89 (1990).

    MathSciNet  MATH  Google Scholar 

  10. K. B. Lee, J. K. Shin and Y. Shoji, “Free actions of finite abelian groups on the 3-Torus,” Topology Appl. 53 (2), 153–175 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Y. Ha, J. H. Jo, S. W. Kim, and J. B. Lee, “Classification of Free Actions of Finite Groups on the 3-Torus,” Topology Appl. 121 (3), 469–507 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Olczykowski and A. Sitarz, On spectral action over Bieberbach manifolds, Acta Phys. Polon. B 42 (6), 1189–1198 (2011).

    Article  MathSciNet  Google Scholar 

  13. M. Sadowski, Topological and Affine Structure of Complete Flat Manifolds, arXiv:math/0502449.

  14. H. Takai, On a Duality for Crossed Products of C*-Algebras, J. Functional Analysis 19 (1975), 25–39.

  15. S. Walters, Projective Modules over the Non-Commutative Sphere, J. London Math. Soc. 51 (3), 589–602 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Walters, Chern characters of Fourier modules, Canad. J. Math. 52 (3), 633–672 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Walters, K-theory of non-commutative spheres arising from the Fourier automorphism, Canad. J. Math. 53 (3), 631–672 (2001).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. Olczykowski.

Additional information

Supported by a grant from The John Templeton Foundation.

Supported by MNII grant 189/6.PRUE/2007/7 and NCN grant 2011/01/B/ST1/06474.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Olczykowski, P., Sitarz, A. K-theory of noncommutative Bieberbach manifolds. Russ. J. Math. Phys. 22, 389–399 (2015). https://doi.org/10.1134/S1061920815030097

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920815030097

Keywords

Navigation