Geometriae Dedicata

, Volume 154, Issue 1, pp 93–101 | Cite as

The \({\bar\mu}\)-invariant of Seifert fibered homology spheres and the Dirac operator

  • Daniel Ruberman
  • Nikolai Saveliev
Original Paper


We derive a formula for the \({\bar\mu}\)-invariant of a Seifert fibered homology sphere in terms of the η-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.


Dirac operator Seifert fibered homology sphere \({\bar\mu}\)-invariant η-invariant Seiberg–Witten equations 

Mathematics Subject Classification (2000)

57M27 57R57 11F20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Atiyah M., Patodi V., Singer I.: Spectral asymmetry and Riemannian geometry: I. Math. Proc. Camb. Phil. Soc. 77, 43–69 (1975)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Collin O., Saveliev N.: Equivariant Casson invariants via gauge theory. J. Reine. Angew. Math. 541, 143–169 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Frøyshov K.: The Seiberg–Witten equations and four-manifolds with boundary. Math. Res. Lett. 3, 373–390 (1996)MathSciNetGoogle Scholar
  4. 4.
    Fukumoto Y., Furuta M.: Homology 3-spheres bounding acyclic 4-manifolds. Math. Res. Lett. 7, 757–766 (2000)MathSciNetMATHGoogle Scholar
  5. 5.
    Hirzebruch, F., Zagier, D.: The Atiyah–Singer theorem and elementary number theory. Publish or Perish (1974)Google Scholar
  6. 6.
    Kronheimer P.: A Torelli-type theorem for gravitational instantons. J. Differential Geom. 29, 685–697 (1989)MathSciNetMATHGoogle Scholar
  7. 7.
    Mrowka, T., Ruberman, D., Saveliev, N.: Seiberg–Witten equations, end-periodic Dirac operators, and a lift of Rohlin’s invariant. Preprint arXiv:0905.4319 [math.GT]Google Scholar
  8. 8.
    Neumann, W.: An invariant of plumbed homology spheres. Topology Symposium, Siegen1979, Lecture Notes in Mathematics, vol. 788, pp. 125–144. Springer, Berlin (1980)Google Scholar
  9. 9.
    Neumann, W., Raymond, F.: Seifert manifolds, plumbing, μ-invariant and orientation reversing maps. In: Algebraic and geometric topology, Santa Barbara, Calif. (1977) Lecture Notes in Mathematics, vol. 664, pp. 163–196. Springer, Berlin (1978)Google Scholar
  10. 10.
    Nicolaescu L.: Eta invariants of Dirac operators on circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg–Witten moduli spaces. Israel J. Math. 114, 61–123 (1999)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Nicolaescu L.: Finite energy Seiberg–Witten moduli spaces on 4-manifolds bounding Seifert fibrations. Comm. Anal. Geom. 8, 1027–1096 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Nicolaescu L.: Lattice points inside rational simplices and the Casson invariant of Brieskorn spheres. Geom. Ded. 88, 37–53 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Rademacher H.: Some remarks on certain generalized Dedekind sums. Acta. Arith. 9, 97–105 (1964)MathSciNetGoogle Scholar
  14. 14.
    Ruberman D., Saveliev N.: Rohlin’s invariant and gauge theory. II. Mapping tori. Geom. Topol. 8, 35–76 (2004)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Saveliev N.: Invariants for Homology 3-spheres, vol. 140 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2002)Google Scholar
  16. 16.
    Saveliev N.: Floer homology of Brieskorn homology spheres. J. Differ. Geom. 53, 15–87 (1999)MathSciNetMATHGoogle Scholar
  17. 17.
    Saveliev N.: Fukumoto-Furuta invariants of plumbed homology 3-spheres. Pacific. J. Math. 205, 465–490 (2002)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Scott P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)MATHCrossRefGoogle Scholar
  19. 19.
    Siebenmann, L.: On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology 3-spheres. Topology Symposium, Siegen 1979, pp. 172–222. Lecture Notes in Mathematics, vol. 788, Springer, Berlin (1980)Google Scholar
  20. 20.
    Stipsicz A.: On the \({\bar\mu}\)-invariant of rational surface singularities. Proc. Am. Math. Soc. 136, 3815–3823 (2008)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ue, M.: The Fukumoto-Furuta and the Ozsvath-Szabo invariants for spherical 3-manifolds. Algebraic topology—old and new, pp. 121–139, Banach Center Publ. 85 Polish Acad. Sci. Inst. Math.,Warsaw (2009)Google Scholar
  22. 22.
    Ue M.: The Neumann–Siebenmann invariant and Seifert surgery. Math. Z. 250, 475–493 (2005)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics, MS 050Brandeis UniversityWalthamUSA
  2. 2.Department of MathematicsUniversity of MiamiCoral GablesUSA

Personalised recommendations