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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
DOI: https://doi.org/10.1090/S0002-9947-99-02304-1
Published electronically: January 27, 1999
MathSciNet review: 1603898
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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.


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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: https://doi.org/10.1090/S0002-9947-99-02304-1
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