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Seiberg-Witten Invariants of Lens Spaces

Published online by Cambridge University Press:  20 November 2018

Liviu I. Nicolaescu
Liviu I. Nicolaescu*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, USA website: http://www.nd.edu/∼lnicolae/ e-mail: [email protected]
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Abstract

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We show that the Seiberg-Witten invariants of a lens space determine and are determined by its Casson-Walker invariant and its Reidemeister-Turaev torsion.

Keywords

58D2757Q1057R1557R1953C2053C25lens spacesSeifert manifoldsSeiberg-Witten invariantsCasson-Walker invariantReidemeister torsioneta invariantsDedekind-Rademacher sums
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 4 , 01 August 2001 , pp. 780 - 808
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Blau, M. and Thompson, G., On the relationship between the Rozanski-Witten and the 3-dimensional Seiberg-Witten invariants. hep-th/0006244Google Scholar
[2] Chen, W., Casson invariant and Seiberg-Witten gauge theory. Turkish J.Math. 21(1997), 61–81.Google Scholar
[3] Chen, W., Dehn surgery formula for Seiberg-Witten invariants of homology 3-spheres. dg-ga/9708006Google Scholar
[4] Fintushel, R. and Stern, R., Instanton homology of Seifert fibered homology three spheres. Proc. London Math. Soc. 61(1990), 109–137.Google Scholar
[5] Furuta, M. and Steer, B., Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points. Adv. in Math. 96(1992), 38–102.Google Scholar
[6] Hirzebruch, F., Neumann, W. D. and Koh, S. S., DifferentiableManifolds and Quadratic Forms. Lect. Notes in Pure and Appl. Math. 4, Marcel Dekker, 1971.Google Scholar
[7] Hirzebruch, F. and Zagier, D., The Atiyah-Singer Index Theorem and Elementary Number Theory. Math. Lect. Series 3, Publish or Perish Inc., Boston, 1974.Google Scholar
[8] Jankins, M. and Neumann, W. D., Lectures on Seifert Manifolds. Brandeis Lecture Notes, 1983.Google Scholar
[9] Lescop, C., Global Surgery Formula for the Casson-Walker Invariant. Annals of Math. Studies 140, Princeton University Press, 1996.Google Scholar
[10] Lim, Y., Seiberg-Witten invariants for 3-manifolds in the case b 1 = 0 or 1 . Pacific J. Math. 195(2000), 179–204.Google Scholar
[11] Lim, Y., The equivalence of Seiberg-Witten and Casson invariants for homology 3-spheres. Math. Res. Letters 6(1999), 631–644.Google Scholar
[12] Marcolli, M. and Wang, B. L., Equivariant Seiberg-Witten-Floer homology. dg-ga/9606003Google Scholar
[13] Marcolli, M. and Wang, B. L., Exact triangles in monopole homology and the Casson-Walker invariant. math.DG/0101127Google Scholar
[14] Meng, G. and Taubes, C. H., = Milnor torsion. Math. Res. Letters 3(1996), 661–674.Google Scholar
[15] Milnor, J., Whitehead torsion. Bull. Amer.Math. Soc. 72(1966), 358–426.Google Scholar
[16] Neumann, W. D. and Raymond, F., Seifert manifolds, plumbing, μ-invariant and orientation reversing maps. In: Lecture Notes in Math. 644, 161–195.Google Scholar
[17] Nicolaescu, L. I., Adiabatic limits of the Seiberg-Witten equations on Seifert manifolds. Comm. Anal. Geom. 6(1998), 301–362.Google Scholar
[18] Nicolaescu, L. I., Eta invariants of Dirac operators on circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces. Israel. J. Math. 114(1999), 61–123.Google Scholar
[19] Nicolaescu, L. I., Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations. Comm. Anal. Geom. 8(2000), 1027–1096.Google Scholar
[20] Nicolaescu, L. I., Lattice points inside rational simplices and the Casson invariant of Brieskorn spheres. Geom. Dedicata, to appear.Google Scholar
[21] Nicolaescu, L. I., Seiberg-Witten invariants of rational homology spheres. math.GJ/0103020Google Scholar
[22] Nicolaescu, L. I., Reidemeister Torsion. notes available at: http://www.nd.edu/Ĩnicolae/ Google Scholar
[23] Orlik, P., Seifert Manifolds. Lect. Motes in Math. 291, Springer-Verlag, 1972.Google Scholar
[24] Ouyang, M., Geometric invariants for Seifert fibered 3-manifolds. Trans. Amer. Math. Soc. 346(1994), 641–659.Google Scholar
[25] Rademacher, H., Some remarks on certain generalized Dedekind sums. Acta Arith. 9(1964), 97–105.Google Scholar
[26] Rademacher, H. and Grosswald, E., Dedekind Sums. The Carus Math. Monographs, MAA, 1972.Google Scholar
[27] von Randow, R., Zür Topologie von dreidimensionalen Baummanigfatigkeiten. Bonner Math. Schriften 14(1962).Google Scholar
[28] Scott, P., The geometries of 3-manifolds. Bull. London. Math. Soc. 15(1983), 401–487.Google Scholar
[29] Turaev, V. G., Euler structures, nonsingular vector fields and torsions of Reidemeister type. Izv. Akad. Nauk. USSR 53(1989); English Transl. Math. USSR-Izv. 34(1990), 627–662.Google Scholar
[30] Turaev, V. G., Torsion invariants of spin c structures on 3-manifolds. Math. Res. Letters 4(1997), 679–695.Google Scholar
[31] Walker, K., An Extension of Casson's Invariant. Annals of Math. Studies 126, Princeton University Press, 1992.Google Scholar