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

Author(s): Florian Deloup
Journal: Trans. Amer. Math. Soc. 351 (1999), 1895-1918.
MSC (1991): Primary 11E81, 57N10
Posted: January 27, 1999
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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:

[Be]
R. Bellman, A brief introduction to theta functions, Holt, Rinehart & Winston, New York, 1961. MR 23:A2556

[BE]
B. C. Berndt and R. J. Evans, The determination of Gauss sums, Bull. of the Amer. Math. Soc. 5, n. 2 (1981), 107-129. MR 82h:10051

[Bl]
F. van der Blij, An invariant of quadratic forms mod 8, Indag. Math. 21 (1959), 291-293. MR 21:7183

[Br]
H. Braun, Geschlechter quadratischer Formen, Journal für Math., Bd 182, H. 1. (1940), 32-49. MR 2:36f

[BM]
G. Brumfield and J. Morgan, Quadratic functions, the index modulo $8$ and a ${\mathbf{Z}}/4{\mathbf{Z}}$-Hirzebruch formula, Topology 12 (1973), 105-122. MR 48:3059

[Ch]
K. Chandrasekharan, Elliptic functions, Grundlehren der mathematischen Wissenschften 281, Berlin, Heidelberg, New York, Springer Verlag, 1985. MR 87e:11058

[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]
A. H. Durfee, Bilinear and quadratic forms on torsion modules, Adv. in Math. 25 (1977), 133-164. MR 58:506

[Fr]
A. Fröhlich, Hermitian and quadratic forms over rings with involution, Quart. J. Math. Oxford 20 (1969), 297-317. MR 40:5642

[KK]
A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on $3$-manifolds, Math. Ann. 253 (1980), 29-42. MR 82b:57007

[Ki1]
R. Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978), 35-56. MR 57:7605

[Ki2]
R. Kirby, The topology of 4-manifolds, Lectures Notes in Mathematics 1374, Berlin, New York, Springer Verlag, 1989. MR 90j:57012

[Kr]
A. Krazer, Zur Theorie der mehrfachen Gaußschen Summen, H. Weber Festschrift, Leipzig (1912), s. 181.

[Ky]
R. Kyle, Branched covering spaces and the quadratic forms of links, Ann. of Math. 59 (1954), 539-548. MR 15:979a

[La]
J. Lannes, Formes quadratiques d'enlacement sur l'anneau des entiers d'un corps de nombres, Ann. Sci. Ecole Norm. Sup., 4ème série 8 (1975), 535-579. MR 54:231

[MH]
J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergebnisse der Math. 73, Berlin, Heidelberg, New York, 1973. MR 58:22129

[MPR]
J. Mattes, M. Polyak, N. Reshetikhin, On invariants of 3-manifolds derived from abelian groups, in Quantum Topology, ed. L. Kauffman, R. Baadhio, World Scientific, 1993. MR 95e:57031

[MOO]
H. Murakami, T. Ohtsuki, M. Okada, Invariants of three-manifolds derived from linking matrices of framed links, Osaka J. Math. 29 (1992), 545-572. MR 93h:57013

[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]
V. Rokhlin in A la recherche de la topologie perdue, Progress in Mathematics, vol. 62, ed. L. Guillou, A. Marin, Birkhäuser, 1986. MR 88k:57002

[Rol]
D. Rolfsen, Knots and Links, Publish or Perish, Inc, Berkeley, 1976. MR 58:24236

[Sa]
C.-H. Sah, Symmetric bilinear forms and quadratic forms, J. Algebra 20 (1972), 144-160. MR 45:3448

[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, Colloque sur les formes quadratiques (1975, Montpellier), Bull. Soc. Math. France, mémoire 48 (1976), 103-115. MR 58:27762

[Tu1]
V. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter, Studies in Math., 1994. MR 95k:57014

[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 28:133

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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
Email: deloup@math.u-strasbg.fr

DOI: 10.1090/S0002-9947-99-02304-1
PII: S 0002-9947(99)02304-1
Keywords: Reciprocity, quadratic form, Gauss sum, Witt group, manifold, linking form, modular category.
Received by editor(s): April 22, 1997
Posted: January 27, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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