Mathematische Annalen

, Volume 299, Issue 1, pp 231–267 | Cite as

Dedekind sums, μ-invariants and the signature cocycle

  • Robion Kirby
  • Paul Melvin
Article

Mathematics Subject Classification (1991)

11F20 57MR 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ap]
    Apostol, T.M.: Modular functions and Dirichlet series in number theory. (Grad. Texts Math., vol 41) Berlin Heidelberg New York: Springer 1976Google Scholar
  2. [At1]
    Atiyah, M.F.: The logarithm of the Dedekind η-function. Math. Ann.278, 335–380 (1987)Google Scholar
  3. [At2]
    Atiyah, M.F.: On framings of 3-manifolds. Topology29, 1–7 (1990)Google Scholar
  4. [Bri]
    Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten. Invent. Math.2, 1–14 (1966)Google Scholar
  5. [Bro1]
    Brown, E.H.: Generalization of the Kervaire invariant. Ann. Math.95, 368–383 (1972)Google Scholar
  6. [Bro2]
    Brown, K.S.: Cohomology of groups. Berlin Heidelberg New York: Springer 1982Google Scholar
  7. [CG]
    Casson, A.J., Gordon, C.McA.: Cobordism of classical knots. In: Guillou, L., Marin, A. (eds.) A la recherche de la topologie perdue (Prog. Math., vol. 62, pp. 181–200) Boston Basel Stuttgart: Birkhäuser 1986Google Scholar
  8. [D]
    Dedekind, R.: Erläuterungen zu zwei Fragmenten von Riemann (1877). In: Fricke, R. et al. (eds.) Dedekind's Gesammelte Math. Werke I, pp. 159–173. Braunschweig: Vieweg 1930Google Scholar
  9. [FG]
    Freed, D.S., Gompf, R.E.: Computer calculation of Witten's 3-manifold invariant. Commun. Math. Phys.141, 79–117 (1991)Google Scholar
  10. [Garx]
    Garoufalidis, S.: Relations among 3-manifold invariants. (Preprint)Google Scholar
  11. [Gui]
    Guillemin, V., Sternberg, S.: Geometric asymptotics. (Math. Surv., vol. 14) Providence: Am. Math. Soc. 1977Google Scholar
  12. [Gun]
    Gunning, R.C.: Lectures on modular forms. (Ann. Math. Stud., vol. 48) Princeton Princeton University Press 1962Google Scholar
  13. [HW]
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: Clarendon 1938Google Scholar
  14. [Hic]
    Hickerson, D.: Continued fractions and density results. J. Reine Angew. Math.290, 113–116 (1977)Google Scholar
  15. [Hir1]
    Hirzebruch, F.: Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen. Math. Ann.126, 1–22 (1953)Google Scholar
  16. [Hir2]
    Hirzebruch, F.: The signature theorem: reminiscences and recreation. In: Prospects in Math. (Ann. Math. Stud., vol. 70, pp. 3–31) Princeton: Princeton University Press 1971Google Scholar
  17. [Hir3]
    Hirzebruch, F.: Hilbert modular surfaces. Enseign. Math.19, 183–281 (1973)Google Scholar
  18. [HirZ]
    Hirzebruch, F., Zagier, D.: The Atiyah-Singer theorem and elementary number theory. Berkeley: Publish or Perish 1974Google Scholar
  19. [J]
    Jeffrey, L.C.: Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation. Commun. Math. Phys.147, 563–604 (1992)Google Scholar
  20. [K1]
    Kirby, R.C.: A calculus for framed links inS 3. Invent. Math.45, 35–56 (1978)Google Scholar
  21. [K2]
    Kirby, R.C.: The topology of 4-manifolds. (Lect. Notes Math., vol. 1374) Berlin Heidelberg New York: Springer 1989Google Scholar
  22. [KM1]
    Kirby, R., Melvin, P.: The 3-manifold invariants of Witten and Reshetikhin-Turaev forsl(2, C). Invent. Math.105, 473–545 (1991)Google Scholar
  23. [KM2]
    Kirby, R., Melvin, P.: Quantum invariants of lens spaces and a Dehn surgery formula. Abstracts Am. Math. Soc.12, 435 (1991)Google Scholar
  24. [Kn]
    Knopp, M.I.: Modular functions in analytic number theory. Chicago: Markham 1970Google Scholar
  25. [L]
    Litherland, R.A.: Signatures of iterated torus knots. In: Fenn, R. (ed.) Topology of lowdimensional manifolds. (Lect. Notes Math., vol. 722, pp. 71–84) Berlin Heidelberg New York: Springer 1979Google Scholar
  26. [Mag]
    Magnus, W.: Noneuclidean tesselations and their groups. (Pure and Applied Math. Ser., vol. 61) New York London: Academic Press 1974Google Scholar
  27. [Mel1]
    Melvin, P.: On 4-manifolds with singular torus actions. Math. Ann.256, 255–276 (1981)Google Scholar
  28. [Mel2]
    Melvin, P.: Tori in the diffeomorphism groups of simply connected 4-manifolds. Math. Proc. Camb. Philos. Soc.91, 305–314 (1982)Google Scholar
  29. [MelK]
    Melvin, P., Kazez, W.: 3-dimensional bordism. Mich. Math. J.36, 251–260 (1989)Google Scholar
  30. [Mey]
    Meyer, W.: Die Signatur von Flächenbündeln. Math. Ann.201, 239–264 (1973)Google Scholar
  31. [MeyS]
    Meyer, W., Sczech, R.: Über eine topologische und zahlentheoretische Anwendung von Hirzebruchs Spitzenauflösung. Math. Ann.240, 69–96 (1979)Google Scholar
  32. [Mor]
    Mordell, L.J.: Lattice points in a tetrahedron and generalized Dedekind sums. J. Indian Math. Soc.15, 41–46 (1951)Google Scholar
  33. [NR]
    Neumann, W.D., Raymond, F.: Seifert manifolds, plumbing, μ-invariant and orientation reversing maps. In: Millett, K.C. (ed.) Algebraic and geometric topology. (Lect. Notes Math., vol. 664, pp. 163–196) Berlin Heidelberg New York: Springer 1978Google Scholar
  34. [O]
    Orlik, P.: Seifert manifolds. (Lect. Notes Math., vol. 291) Berlin Heidelberg New York: Springer 1972Google Scholar
  35. [Rad1]
    Rademacher, H.A.: Zur Theorie der Modulfunktionen. J. Reine Angew. Math.167, 312–366 (1931)Google Scholar
  36. [Rad2]
    Rademacher, H.A.: Lectures on analytic number theory. Notes. Bombay: Tata Institute of Fundamental Research 1954–1955Google Scholar
  37. [RadG]
    Rademacher, H., Grosswald, E.: Dedekind sums. (Carus. Math. Monogr., vol. 16) Math. Assoc. Am. 1972Google Scholar
  38. [ResT]
    Reshetikhin, N.Yu., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.103, 547–597 (1991)Google Scholar
  39. [Rob]
    Robertello, R.A.: An invariant of knot cobordism. Commun. Pure Appl. Math.18, 543–555 (1965)Google Scholar
  40. [Rol]
    Rolfsen, D.: Knots and links. Berkeley: Publish or Perish 1976Google Scholar
  41. [Rou]
    Rourke, C.P.: A new proof that Ω3 is zero. J. Lond. Math. Soc.31, 373–376 (1985)Google Scholar
  42. [Sc]
    Sczech, R.: Die α-Invariante von gewissen Aktionen aufT 2-Bündeln und ihr Zusammenhang mit der Zahlentheorie. Diplomarbeit Bonn (1975)Google Scholar
  43. [Se]
    Serre, J.-P.: A course in arithmetic. Berlin Heidelberg New York: Springer 1973Google Scholar
  44. [V]
    Viro, O.Ja.: Branched coverings of manifolds with boundary and link invariants. I. Math. USSR, Isv.7, 1239–1255 (1973)Google Scholar
  45. [Wa]
    Walker, K.: An extension of Casson's invariant (Ann. Math. Stud., vol. 126) Princeton: Princeton University Press 1992Google Scholar
  46. [Wi]
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989)Google Scholar
  47. [Wo]
    Woodard, M.R.: The Rohlin invariant of surgered, sewn link exteriors. Proc. Am. Math. Soc.112, 211–221 (1991)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Robion Kirby
    • 1
  • Paul Melvin
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsBryn Mawr CollegeBryn MawrUSA

Personalised recommendations