zbMATH — the first resource for mathematics

Dedekind sums, \(\mu\)-invariants and the signature cocycle. (English) Zbl 0809.11027
This paper gives a geometric formulation of the Rademacher \(\varphi\)- function and of Dedekind sums in terms of the action of the modular group on the hyperbolic plane. This approach leads to simple continued fraction formulas involving signatures, which illuminate the appearance of Dedekind sums in many signature related formulas in topology. Several of these are discussed in the paper, including formulas for the signature defects and \(\mu\)-invariants of lens spaces and torus bundles over the circle, and for the well known signature cocycle on \(\text{SL} (2,\mathbb{Z})\). Some consequences of this point of view include (1) the existence of naturally defined integer lifts of the \(\mu\)-invariants of lens spaces (related also to their Brown invariants), (2) new formulas for \(p\)-signatures of knots, and (3) a simple formula for the signature cocycle.
Reviewer: R.Kirby (Berkeley)

11F20 Dedekind eta function, Dedekind sums
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
Full Text: DOI EuDML
[1] Apostol, T.M.: Modular functions and Dirichlet series in number theory. (Grad. Texts Math., vol 41) Berlin Heidelberg New York: Springer 1976 · Zbl 0332.10017
[2] Atiyah, M.F.: The logarithm of the Dedekind ?-function. Math. Ann.278, 335-380 (1987) · Zbl 0648.58035
[3] Atiyah, M.F.: On framings of 3-manifolds. Topology29, 1-7 (1990) · Zbl 0716.57011
[4] Brieskorn, E.: Beispiele zur Differentialtopologie von Singularit?ten. Invent. Math.2, 1-14 (1966) · Zbl 0145.17804
[5] Brown, E.H.: Generalization of the Kervaire invariant. Ann. Math.95, 368-383 (1972) · Zbl 0241.57014
[6] Brown, K.S.: Cohomology of groups. Berlin Heidelberg New York: Springer 1982 · Zbl 0584.20036
[7] 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: Birkh?user 1986
[8] Dedekind, R.: Erl?uterungen zu zwei Fragmenten von Riemann (1877). In: Fricke, R. et al. (eds.) Dedekind’s Gesammelte Math. Werke I, pp. 159-173. Braunschweig: Vieweg 1930
[9] Freed, D.S., Gompf, R.E.: Computer calculation of Witten’s 3-manifold invariant. Commun. Math. Phys.141, 79-117 (1991) · Zbl 0739.53065
[10] Garoufalidis, S.: Relations among 3-manifold invariants. (Preprint)
[11] Guillemin, V., Sternberg, S.: Geometric asymptotics. (Math. Surv., vol. 14) Providence: Am. Math. Soc. 1977 · Zbl 0364.53011
[12] Gunning, R.C.: Lectures on modular forms. (Ann. Math. Stud., vol. 48) Princeton Princeton University Press 1962 · Zbl 0178.42901
[13] Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: Clarendon 1938 · Zbl 0020.29201
[14] Hickerson, D.: Continued fractions and density results. J. Reine Angew. Math.290, 113-116 (1977) · Zbl 0341.10012
[15] Hirzebruch, F.: ?ber vierdimensionale Riemannsche Fl?chen mehrdeutiger analytischer Funktionen von zwei komplexen Ver?nderlichen. Math. Ann.126, 1-22 (1953) · Zbl 0093.27605
[16] Hirzebruch, F.: The signature theorem: reminiscences and recreation. In: Prospects in Math. (Ann. Math. Stud., vol. 70, pp. 3-31) Princeton: Princeton University Press 1971 · Zbl 0252.58009
[17] Hirzebruch, F.: Hilbert modular surfaces. Enseign. Math.19, 183-281 (1973) · Zbl 0285.14007
[18] Hirzebruch, F., Zagier, D.: The Atiyah-Singer theorem and elementary number theory. Berkeley: Publish or Perish 1974 · Zbl 0288.10001
[19] Jeffrey, L.C.: Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation. Commun. Math. Phys.147, 563-604 (1992) · Zbl 0755.53054
[20] Kirby, R.C.: A calculus for framed links inS 3. Invent. Math.45, 35-56 (1978) · Zbl 0377.55001
[21] Kirby, R.C.: The topology of 4-manifolds. (Lect. Notes Math., vol. 1374) Berlin Heidelberg New York: Springer 1989 · Zbl 0668.57001
[22] Kirby, R., Melvin, P.: The 3-manifold invariants of Witten and Reshetikhin-Turaev forsl(2, C). Invent. Math.105, 473-545 (1991) · Zbl 0745.57006
[23] Kirby, R., Melvin, P.: Quantum invariants of lens spaces and a Dehn surgery formula. Abstracts Am. Math. Soc.12, 435 (1991)
[24] Knopp, M.I.: Modular functions in analytic number theory. Chicago: Markham 1970 · Zbl 0259.10001
[25] 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 1979 · Zbl 0412.57002
[26] Magnus, W.: Noneuclidean tesselations and their groups. (Pure and Applied Math. Ser., vol. 61) New York London: Academic Press 1974 · Zbl 0293.50002
[27] Melvin, P.: On 4-manifolds with singular torus actions. Math. Ann.256, 255-276 (1981) · Zbl 0453.57007
[28] Melvin, P.: Tori in the diffeomorphism groups of simply connected 4-manifolds. Math. Proc. Camb. Philos. Soc.91, 305-314 (1982) · Zbl 0486.57016
[29] Melvin, P., Kazez, W.: 3-dimensional bordism. Mich. Math. J.36, 251-260 (1989) · Zbl 0693.57012
[30] Meyer, W.: Die Signatur von Fl?chenb?ndeln. Math. Ann.201, 239-264 (1973) · Zbl 0245.55017
[31] Meyer, W., Sczech, R.: ?ber eine topologische und zahlentheoretische Anwendung von Hirzebruchs Spitzenaufl?sung. Math. Ann.240, 69-96 (1979) · Zbl 0392.14007
[32] Mordell, L.J.: Lattice points in a tetrahedron and generalized Dedekind sums. J. Indian Math. Soc.15, 41-46 (1951) · Zbl 0043.05101
[33] Neumann, W.D., Raymond, F.: Seifert manifolds, plumbing, ?-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 1978 · Zbl 0401.57018
[34] Orlik, P.: Seifert manifolds. (Lect. Notes Math., vol. 291) Berlin Heidelberg New York: Springer 1972 · Zbl 0263.57001
[35] Rademacher, H.A.: Zur Theorie der Modulfunktionen. J. Reine Angew. Math.167, 312-366 (1931) · Zbl 0003.21501
[36] Rademacher, H.A.: Lectures on analytic number theory. Notes. Bombay: Tata Institute of Fundamental Research 1954-1955
[37] Rademacher, H., Grosswald, E.: Dedekind sums. (Carus. Math. Monogr., vol. 16) Math. Assoc. Am. 1972 · Zbl 0251.10020
[38] Reshetikhin, N.Yu., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.103, 547-597 (1991) · Zbl 0725.57007
[39] Robertello, R.A.: An invariant of knot cobordism. Commun. Pure Appl. Math.18, 543-555 (1965) · Zbl 0151.32501
[40] Rolfsen, D.: Knots and links. Berkeley: Publish or Perish 1976 · Zbl 0339.55004
[41] Rourke, C.P.: A new proof that ?3 is zero. J. Lond. Math. Soc.31, 373-376 (1985) · Zbl 0585.57012
[42] Sczech, R.: Die ?-Invariante von gewissen Aktionen aufT 2-B?ndeln und ihr Zusammenhang mit der Zahlentheorie. Diplomarbeit Bonn (1975)
[43] Serre, J.-P.: A course in arithmetic. Berlin Heidelberg New York: Springer 1973 · Zbl 0256.12001
[44] Viro, O.Ja.: Branched coverings of manifolds with boundary and link invariants. I. Math. USSR, Isv.7, 1239-1255 (1973) · Zbl 0295.55002
[45] Walker, K.: An extension of Casson’s invariant (Ann. Math. Stud., vol. 126) Princeton: Princeton University Press 1992 · Zbl 0752.57011
[46] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351-399 (1989) · Zbl 0667.57005
[47] Woodard, M.R.: The Rohlin invariant of surgered, sewn link exteriors. Proc. Am. Math. Soc.112, 211-221 (1991) · Zbl 0728.57014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.