Skip to main content
SpringerLink
Log in

Eta invariants for flat manifolds

  • Original Paper
  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Using a formula from Donnelly (Indiana Univ Math J 27(6):889–918, 1978), we prove that for a family of seven dimensional flat manifolds with cyclic holonomy groups the η invariant of the signature operator is an integer number. We also present an infinite family of flat manifolds with integral η invariant. The main motivation is a paper of Long and Reid (Geom Topol 4:171–178, 2000).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Cambridge Philos. Soc. 77, 43–69 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry II. Math. Proc. Cambridge Philos. Soc. 78, 405–432 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brown H., Bülow R., Neubüser J., Wondratschek W., Zassenhaus H.: Crystallographic groups of four-dimensional space. Wiley, New York (1978)

    MATH  Google Scholar 

  4. Charlap L.S.: Bieberbach Groups and Flat Manifolds, Universitext. Springer-Verlag, New York (1986)

    Book  Google Scholar 

  5. Donnelly H.: Eta invariants for G-spaces. Indiana Univ. Math. J. 27(6), 889–918 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gilkey P.B., Miatello R.J., Podestá R.A.: The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order. Ann. Global Anal. Geom. 37(3), 275–306 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hiller H.: The crystallographic restriction in higher dimensions. Acta Cryst. A41, 541–544 (1985)

    MathSciNet  Google Scholar 

  8. Hiller H., Sah C.H.: Holonomy of flat manifolds with b 1 = 0. Quart. J. Math. Oxford Ser. (2) 37, 177–187 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Long D.D., Reid A.W.: On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol. 4, 171–178 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lutowski, R.: Seven dimensional flat manifolds with cyclic holonomy group. Gdańsk, (2010). arXiv:1101.2633 (preprint)

  11. Lutowski, R.: A list of 7-dimensional Bieberbach groups with cyclic holonomy. http://rlutowsk.mat.ug.edu.pl/flat7cyclic/

  12. Miatello R.J., Podestá R.A.: The spectrum of twisted Dirac operators on compact flat manifolds. Trans. Amer. Math. Soc. 358(10), 4569–4603 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Opgenorth, J., Plesken, W., Schulz, T.: CARAT, Crystallographic Algorithms and Tables. http://wwwb.math.rwth-aachen.de/CARAT/

  14. Pfäffle F.: The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35(4), 367–385 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rossetti J.P., Szczepański A.: Generalized Hantzsche-Wendt flat manifolds. Rev. Mat. Iberoamericana 21(3), 1053–1070 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sadowski M., Szczepański A.: Flat manifolds, harmonic spinors, and eta invariants. Adv. Geom. 6(2), 287–300 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Serre J.-P.: Linear Representations of Finite Groups. Springer-Verlag, Berlin (1977)

    MATH  Google Scholar 

  18. Szczepański, A.: Geometry of the crystallographic groups, book in preparation available on web http://www.mat.ug.edu.pl/aszczepa

  19. Wolf J.: Spaces of constant curvature. MacGraw Hill, New York (1967)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952, Gdańsk, Poland

    A. Szczepański

Authors
  1. A. Szczepański
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Szczepański.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Szczepański, A. Eta invariants for flat manifolds. Ann Glob Anal Geom 41, 125–138 (2012). https://doi.org/10.1007/s10455-011-9274-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-011-9274-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation