Annals of Global Analysis and Geometry

, Volume 37, Issue 3, pp 275–306 | Cite as

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

  • Peter B. Gilkey
  • Roberto J. MiatelloEmail author
  • Ricardo A. Podestá


We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \({\mathbb{Z}_p}\), where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.


Flat manifolds Eta invariant Equivariant bordism 

Mathematics Subject Classification (2000)



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  1. 1.
    Ambrose W., Singer I.M.: A theorem on holonomy. Trans. Am. Math. Soc. 75, 428–443 (1953)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Anderson D.W., Brown E.H. Jr, Peterson F.P.: Spin cobordism. Bull. Am. Math. Soc. 72, 256–260 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Anderson D.W., Brown E.H. Jr, Peterson F.P.: The structure of the Spin cobordism ring. Ann. Math. 86, 271–298 (1967)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Atiyah M.F., Patodi F.K., Singer I.M.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Phil. Soc. 77, 43–69 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bahri A., Bendersky M., Davis D., Gilkey P.: The complex bordism of groups with periodic cohomology. Trans. Am. Math. Soc. 316, 673–687 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bahri A., Bendersky M., Gilkey P.: The relationship between complex bordism and K-theory for groups with periodic cohomology. Contemp. Math. 96, 19–31 (1989)MathSciNetGoogle Scholar
  7. 7.
    Botvinnik B., Gilkey P., Stolz S.: The Gromov–Lawson–Rosenberg conjecture for groups with periodic cohomology. J. Differ. Geom. 46, 374–405 (1997)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Brown H., Bülow R., Neubüser J., Wondratschok H., Zassenhaus H.: Crystallographic Groups of Four-Dimensional Space. Wiley, New York (1978)zbMATHGoogle Scholar
  9. 9.
    Charlap L.: Compact flat Riemannian manifolds I. Ann. Math. 81, 15–30 (1965)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Charlap L.: Bieberbach Groups and Flat Manifolds. UTM, Springer, New York (1988)Google Scholar
  11. 11.
    Cid C., Schulz T.: Computation of five and six dimensional Bieberbach groups. Exp. Math. 10, 109–115 (2001)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Conway J.H., Rossetti J.P.: Describing the platycosms. Math. Res. Lett. 13, 475–494 (2006)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Friedrich, T.: Dirac operator in Riemannian geometry, Vol. 25. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1997)Google Scholar
  14. 14.
    Gilkey P.B.: The residue of the global η function at the origin. Adv. Math. 40, 290–307 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gilkey P.B.: The Geometry of Spherical Space Form Groups Series in Pure Mathematics, vol. 7. World Scientific, Singapore (1988)Google Scholar
  16. 16.
    Gilkey P.B.: Invariance Theory, the Heat Equation and The Atiyah-Singer Index Theorem, 2nd ed. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)Google Scholar
  17. 17.
    Gilkey P., Leahy J., Park J.H.: Spectral Geometry, Riemannian Submersions, and the Gromov–Lawson Conjecture. CRC Press, Boca Raton (1999)zbMATHGoogle Scholar
  18. 18.
    Hantzsche W., Wendt H.: Dreidimensionale euklidische Raumformen. Math. Ann. 10, 593–611 (1935)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Lawson H.B., Michelsohn M.L.: Spin Geometry. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
  20. 20.
    Lichnerowicz A.: Spineurs harmoniques. C. R. Acad. Sci. Paris 257, 7–9 (1963)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Miatello R.J., Podestá R.A.: Spin structures and spectra of \({\mathbb {Z}_2^k}\)-manifolds. Math. Zeitschrift 247, 319–335 (2004)zbMATHCrossRefGoogle Scholar
  22. 22.
    Miatello R.J., Podestá R.A.: The spectrum of twisted Dirac operators on compact flat manifolds. Trans. Amer. Math. Soc. 358, 4569–4603 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Miatello R.J., Podestá R.A.: Eta invariants and class numbers. Pure Appl. Math. Q. 5, 1–26 (2009)MathSciNetGoogle Scholar
  24. 24.
    Miatello, R.J., Rossetti, J.P.: Spectral properties of flat manifolds. In: Gordon, C.S., Tirao, J.A., Vargas, J., Wolf, J.A. (eds.) New developments in Lie theory and geometry. Sixth Workshop on Lie theory and Geometry, November 13–17 2007, Cruz Chica, Córdoba, Argentina. Contemporary Mathematics, vol. 491, pp. 83–113 (2009)Google Scholar
  25. 25.
    Pfäffle F.: The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35, 367–385 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Reiner I.: Integral representations of cyclic groups of prime order. Proc. Am. Math. Soc. 8, 142–145 (1957)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sadowski M., Szczepanski A.: Flat manifolds, harmonic spinors and eta invariants. Adv. Geom. 6, 287–300 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Seeley R.T.: Complex powers of an elliptic operator. Singular integrals. Proc. Sympos. Pure Math. 10, 288–307 (1967)MathSciNetGoogle Scholar
  29. 29.
    Stolz S.: Simply connected manifolds of positive scalar curvature. Ann. Math. 136, 511–540 (1992)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Thom R.: Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28, 17–86 (1954)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Vasquez A.: Flat Riemannian manifolds. J. Differ. Geom. 4, 367–382 (1970)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Wolf J.: Spaces of Constant Curvature. McGraw-Hill, New York (1967)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Peter B. Gilkey
    • 1
  • Roberto J. Miatello
    • 2
    Email author
  • Ricardo A. Podestá
    • 2
  1. 1.Mathematics DepartmentUniversity of OregonEugeneUSA
  2. 2.FaMAF – CIEM, Universidad Nacional de CórdobaCórdobaArgentina

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