Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Linking forms, reciprocity for Gauss sums and invariants of 3-manifolds

Author: Florian Deloup
Journal: Trans. Amer. Math. Soc. 351 (1999), 1895-1918
MSC (1991): Primary 11E81, 57N10
Published electronically: January 27, 1999
MathSciNet review: 1603898
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study invariants of $3$-manifolds derived from finite abelian groups equipped with quadratic forms. These invariants arise in Turaev's theory of modular categories and generalize those of H. Murakami, T. Ohtsuki and M. Okada. The crucial algebraic tool is a new reciprocity formula for Gauss sums, generalizing classical formulas of Cauchy, Kronecker, Krazer and Siegel. We use this reciprocity formula to give an explicit formula for the invariants and to generalize them to higher dimensions.

References [Enhancements On Off] (What's this?)

  • [Be] Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR 0125252
  • [BE] Bruce C. Berndt and Ronald J. Evans, The determination of Gauss sums, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 107–129. MR 621882, 10.1090/S0273-0979-1981-14930-2
  • [Bl] F. van der Blij, An invariant of quadratic forms mod 8, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 291–293. MR 0108467
  • [Br] Hel Braun, Geschlechter quadratischer Formen, J. Reine Angew. Math. 182 (1940), 32–49 (German). MR 0002351
  • [BM] Gregory W. Brumfiel and John W. Morgan, Quadratic functions, the index modulo 8, and a 𝑍/4-Hirzebruch formula, Topology 12 (1973), 105–122. MR 0324709
  • [Ch] K. Chandrasekharan, Elliptic functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396
  • [Dab] R. Dabrowski, Multivariate Gauss sums, preprint, Columbia University 1995.
  • [Del] F. Deloup, Linking forms, reciprocity for Gauss sums and invariants of $3$-manifolds, prépublication de l'IRMA no. 26, Strasbourg 1996. CMP 98:08
  • [Du] Alan H. Durfee, Bilinear and quadratic forms on torsion modules, Advances in Math. 25 (1977), no. 2, 133–164. MR 0480333
  • [Fr] A. Fröhlich, Hermitian and quadratic forms over rings with involution, Quart. J. Math. Oxford Ser. (2) 20 (1969), 297–317. MR 0252422
  • [KK] Akio Kawauchi and Sadayoshi Kojima, Algebraic classification of linking pairings on 3-manifolds, Math. Ann. 253 (1980), no. 1, 29–42. MR 594531, 10.1007/BF01457818
  • [Ki1] Robion Kirby, A calculus for framed links in 𝑆³, Invent. Math. 45 (1978), no. 1, 35–56. MR 0467753
  • [Ki2] Robion C. Kirby, The topology of 4-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989. MR 1001966
  • [Kr] A. Krazer, Zur Theorie der mehrfachen Gaußschen Summen, H. Weber Festschrift, Leipzig (1912), s. 181.
  • [Ky] R. H. Kyle, Branched covering spaces and the quadratic forms of links, Ann. of Math. (2) 59 (1954), 539–548. MR 0062438
  • [La] J. Lannes, Formes quadratiques d’enlacement sur l’anneau des entiers d’un corps de nombres, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 535–579 (French). MR 0412102
  • [MH] John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. MR 0506372
  • [MPR] Josef Mattes, Michael Polyak, and Nikolai Reshetikhin, On invariants of 3-manifolds derived from abelian groups, Quantum topology, Ser. Knots Everything, vol. 3, World Sci. Publ., River Edge, NJ, 1993, pp. 324–338. MR 1273582, 10.1142/9789812796387_0018
  • [MOO] Hitoshi Murakami, Tomotada Ohtsuki, and Masae Okada, Invariants of three-manifolds derived from linking matrices of framed links, Osaka J. Math. 29 (1992), no. 3, 545–572. MR 1181121
  • [Mu] H. Murakami, Quantum SO(3)-invariants dominate the SU(2)-invariant of Casson and Walker, preprint, Osaka University, 1992.
  • [Oh] T. Ohtsuki, A polynomial invariant of rational homology $3$-spheres, preprint, 1994.
  • [Rok] Lucien Guillou and Alexis Marin (eds.), À la recherche de la topologie perdue, Progress in Mathematics, vol. 62, Birkhäuser Boston, Inc., Boston, MA, 1986 (French). I. Du c\cflex oté de chez Rohlin. II. Le c\cflex oté de Casson. [I. Rokhlin’s way. II. Casson’s way]. MR 900243
  • [Rol] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. MR 0515288
  • [Sa] Chih Han Sah, Symmetric bilinear forms and quadratic forms, J. Algebra 20 (1972), 144–160. MR 0294378
  • [Sc] W. Scharlau, Quadratic and hermitian forms, Heidelberg, New York, Tokyo, Springer-Verlag, 1986.
  • [Si] C. L. Siegel, Uber die analytische Theorie der quadratischen Formen, Ann. Math., 36 (1935), 527.
  • [Sp] T. A. Springer, Caractères quadratiques de groupes abéliens finis et sommes de Gauss, Bull. Soc. Math. France Suppl. Mem. 48 (1976), 103–115 (French). Colloque sur les Formes Quadratiques (Montpellier, 1975). MR 0562091
  • [Tu1] V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
  • [Tu2] V. Turaev, Cohomology rings, linking forms and invariants of spin structures of three-dimensional manifolds, Math. USSR Sbornik, Vol. 48 (1984) No.1.
  • [Tu3] V. Turaev, private conversation, 1995.
  • [Wa] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR 0156890

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11E81, 57N10

Retrieve articles in all journals with MSC (1991): 11E81, 57N10

Additional Information

Florian Deloup
Affiliation: Institut de Recherche en Mathématiques Avancées 7, rue René Descartes 67084 Strasbourg, France
Address at time of publication: Laboratoire de Mathématiques, Emile Picard, Université Paul Sabatier, Toulouse III, 118, route de Narbonne, 31062 Toulouse, France

Keywords: Reciprocity, quadratic form, Gauss sum, Witt group, manifold, linking form, modular category.
Received by editor(s): April 22, 1997
Published electronically: January 27, 1999
Article copyright: © Copyright 1999 American Mathematical Society