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

Transactions of the American Mathematical Society

Volumn 351, Issue 5, 1999, Pages 1895-1918

  Deloup, Florian ^a,b  

^a UNIVERSITÉ LOUIS PASTEUR   (France)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Gauss sum; Linking form; Manifold; Modular category; Quadratic form; Reciprocity; Witt group

Indexed keywords

EID: 22644452312     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-99-02304-1     Document Type: Article

References (31)

1. [Be] R. Bellman, A brief introduction to theta functions, Holt, Rinehart & Winston, New York 1961. MR 23.-A2556
2. [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
3. [Bl] F. van der Blij, An invariant of quadratic forms mod 8, Indag. Math. 21 (1959), 291-293 MR 21:7183
4. [Br] H. Braun, Geschlechter quadratischer Formen, Journal für Math., Bd 182, H. 1. (1940) 32-49. MR 2:36f
5. [BM] G. Brumfield and J. Morgan, Quadratic functions, the index modulo 8 and a Z/4Z Hirzebruch formula, Topology 12 (1973), 105-122. MR 48:3059
6. [Ch] K. Chandrasekharan, Elliptic functions, Grundlehren der mathematischen Wissenschfte 281, Berlin, Heidelberg, New York, Springer Verlag, 1985. MR 87e:11058
7. [Dab] R. Dabrowski, Multivariate Gauss sums, preprint, Columbia University 1995.
8. [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
9. [Du] A. H. Durfee, Bilinear and quadratic forms on torsion modules, Adv. in Math. 25 (1977) 133-164. MR 58:506
10. [Fr] A. Fröhlich, Hermitian and quadratic forms over rings with involution, Quart. J. Math Oxford 20 (1969), 297-317. MR 40:5642
11. [KK] A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on 3-manifolds Math. Ann. 253 (1980), 29-42. MR 82b:57007
12. [Kil] R. Kirby, A calculus for framed links in S3, Invent. Math. 45 (1978), 35-56. MR 57:7605
13. [Ki2] R. Kirby, The topology of 4-manifolds, Lectures Notes in Mathematics 1374, Berlin, Ne York, Springer Verlag, 1989. MR 90j:57012
14. [Kr] A. Krazer, Zur Theorie der mehrfachen Gauschen Summen, H. Weber Festschrift Leipzig (1912), s. 181.
15. [Ky] R. Kyle, Branched covering spaces and the quadratic forms of links, Ann. of Math. 59 (1954), 539-548. MR 15:979a
16. [La] J. Lannes, Formes quadratiques d'enlacement sur l'anneau des entiers d'un corps d nombres, Ann. Sei. Ecole Norm. Sup., 4ème série 8 (1975), 535-579. MR 54:231
17. [MH] J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergebnisse der Math. 73, Berlin Heidelberg, New York, 1973. MR 58:22129
18. [MPR] J. Mattes, M. Polyak, N. Reshetikhin, On invariants of 3-manifolds derived from abelia groups, in Quantum Topology, ed. L. Kauffman, R. Baadhio, World Scientific, 1993. M 95e:57031
19. [MOO] H. Murakami, T. Ohtsuki, M. Okada, Invariants of three-manifolds derived from linkin matrices of framed links, Osaka J. Math. 29 (1992), 545-572. MR 93h:57013
20. [Mu] H. Murakami, Quantum SO(3)-invariants dominate the SU(2)-invariant of Casson an Walker, preprint, Osaka University, 1992
21. [Oh] T. Ohtsuki, A polynomial invariant of rational homology 3-spheres, preprint, 1994.
22. [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
23. [Rol] D. Rolfsen, Knots and Links, Publish or Perish, Inc, Berkeley, 1976. MR 58:24236
24. [Sa] C.-H. Sah, Symmetric bilinear forms and quadratic forms, 3. Algebra 20 (1972), 144-160 MR 45:3448
25. [Sc] W. Scharlau, Quadratic and hermitian forms, Heidelberg, New York, Tokyo, SpringerVerlag, 1986.
26. [Si] C. L. Siegel, Über die analytische Theorie der quadratischen Formen, Ann. Math., 36 (1935), 527.
27. [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
28. [Tul] V. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter, Studies in Math. 1994. MR 95k:57014
29. [Tu2] V. Turaev, Cohomology rings, linking forms and invariants of spin structures of threedimensional manifolds, Math. USSR Sbornik, Vol. 48 (1984) No.l.
30. [Tu3] V. Turaev, private conversation, 1995.
31. [Wa] C.T.C Wall, Quadratic forms on finite groups and related topics, Topology 2 (1963) 281-298. MR 28:133