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Seiberg—Witten on three-manifolds

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Seiberg-Witten Gauge Theory

Part of the book series: Texts and Readings in Mathematics ((TRM,volume 17))

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Abstract

We briefly review some basic notions of 3-manifold topology. We refer the reader to [31] for a brief but informative overview.

I do not think

that I know it well;

but I know not

that I do not know.

Who of us knows that,

he does know that;

but he does not know

that he does not know.

Kena Upaniṣad, 2.2

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Bibliography

  1. S. Akbulut, J.D. McCarthy, Casson’s invariant for oriented homology 3-spheres. An exposition, Princeton University Press, 1990.

    Book  MATH  Google Scholar 

  2. M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry, I.II, III. Math. Proc. Cambridge Phil. Soc. 77 (1975) 43–69. 78 (1975) 405–432. 79 (1976) 71–99.

    Article  MathSciNet  MATH  Google Scholar 

  3. D.M. Austin, P.J. Braam, Equivariant Floer theory and gluing Donaldson polynomials, Topology 35, N.1 (1996) 167–200.

    Article  MathSciNet  MATH  Google Scholar 

  4. D.M. Austin, P.J. Braam, Morse-Bott theory and equivariant cohomology, in The Floer Memorial Volume, Progress in Mathematics, Birkhäuser 1995, 123–183.

    Chapter  Google Scholar 

  5. A. Bertram, M. Thaddeus, On the quantum cohomology of a symmetric product of an algebraic curve, preprint math.AG/9803026.

    Google Scholar 

  6. P.J. Braam, S.K. Donaldson, Floer’s work on instanton homology, knots and surgery, Floer Memorial Volume, Progress in Mathematics, Vol. 133. Birkhäuser 1995.

    MATH  Google Scholar 

  7. A.L. Carey, M. Marcolli, B.L. Wang, Exact triangles in Seiberg-Witten Floer theory, preprint.

    Google Scholar 

  8. A.L. Carey, B.L. Wang, Seiberg-Witten-Floer homology and Gluing formulae, preprint.

    Google Scholar 

  9. A.L. Carey, B.L. Wang, Seiberg-Witten Floer theory and holomorphic curves, preprint.

    Google Scholar 

  10. W. Chen, Casson’s invariant and Seiberg-Witten gauge theory, Turkish J. Math. 21 (1997), no. 1, 61–81.

    MathSciNet  MATH  Google Scholar 

  11. W. Chen, Dehn surgery formula for Seiberg-Witten invariants of homology 3-spheres, preprint, dg-ga/9703009.

    Google Scholar 

  12. S. Dostoglou, D. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. (2) 139 (1994) N.3, 581–640.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Floer, An Instanton-Invariant for 3-Manifolds, Commun. Math. Phys. 118 (1988), 215–240.

    Article  MathSciNet  MATH  Google Scholar 

  14. K.A. Froyshov, The Seiberg-Witten equations and four-manifolds with boundary, Math. Res. Lett. 3 (1996), no. 3, 373–390.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Gabai, Foliations and genera of links, Topology 23 (1984) N.4 381–394.

    Article  MathSciNet  MATH  Google Scholar 

  16. P.B. Kronheimer, Embedded surfaces and gauge theory in three and four dimensions, preprint.

    Google Scholar 

  17. Ch. Lescop, Global surgery formula for the Casson-Walker invariant, Princeton 1996.

    Book  MATH  Google Scholar 

  18. Y. Lim, The equivalence of Seiberg-Witten and Casson invariants for homology 3-spheres, preprint.

    Google Scholar 

  19. Y. Lim, Seiberg-Witten invariants for 3-manifolds in the case b 1 = 0 or 1, preprint.

    Google Scholar 

  20. M. Marcolli, Seiberg-Witten-Floer homology and Heegaard splittings, Internat. J. Math. 7 (1996), no. 5, 671–696.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Marcolli, B.L. Wang, Equivariant Seiberg-Witten Floer homology, preprint.

    Google Scholar 

  22. G. Meng, C. H. Taubes, SW =Milnor torsion, Math. Res. Lett. 3 (1996) N.5, 661–674.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.W. Morgan, T. Mrowka, D. Ruberman, The L 2-moduli space and a vanishing theorem for Donaldson polynomial invariants, International Press, 1994.

    MATH  Google Scholar 

  24. J.W. Morgan, Z. Szabó, C.H. Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996), no. 4, 706–788.

    Article  MathSciNet  MATH  Google Scholar 

  25. T.S. Mrowka, P. Ozsvath, B. Yu, Seiberg-Witten monopoles on Seifert fibered spaces. Comm. Anal. Geom. 5 (1997), no. 4, 685–791.

    Article  MathSciNet  MATH  Google Scholar 

  26. V. Mufioz, B.L. Wang, Seiberg-Witten Floer homology of a surface times a circle, preprint.

    Google Scholar 

  27. W. Neumann, F. Raymond, Seifert manifolds, plumbing, μ-invariant, and orientation reversing maps, in Algebraic and Geometric Topology, Lecture Notes in Math. 664, Springer 1978, 162–194.

    Google Scholar 

  28. W. Neumann, J. Wahl, Casson invariant of links of singularities, Comment. Math. Helvetia 65 (1990) 58–78.

    Article  MathSciNet  MATH  Google Scholar 

  29. L. I. Nicolaescu, Lattice points, Dedekind-Rademacher sums and a conjecture of Kronheimer and Mrowka, preprint, math.DG/9801030.

    Google Scholar 

  30. L. I. Nicolaescu, Eta invariants of Dirac operators on Circle bundles over Riemann surfaces and virtual dimension of finite energy Seiberg-Witten moduli spaces, math.DG/9805046.

    Google Scholar 

  31. Ch. Okonek, A. Teleman, 3-dimensional Seiberg-Witten invariants and non-Kählerian geometry, preprint.

    Google Scholar 

  32. V.V. Prasolov, A.B. Sossinsky, Knots, Links, Braids, and 3-manifolds, AMS 1997.

    MATH  Google Scholar 

  33. M. Schwarz, Morse Homology, Birkhäuser 1993.

    Book  MATH  Google Scholar 

  34. P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983) 401–487.

    Article  MathSciNet  MATH  Google Scholar 

  35. L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric, problems, Ann. of Math. 118 (1983) 525–571.

    Article  MathSciNet  MATH  Google Scholar 

  36. C.H. Taubes, Casson’s Invariant and Gauge Theory, J. Diff. Geom. 31 (1990) 547–599.

    Article  MathSciNet  MATH  Google Scholar 

  37. C.H. Taubes, The stable topology of self-dual moduli spaces, J.Diff.Geom. 29 (1989) 163–230.

    Article  MathSciNet  MATH  Google Scholar 

  38. C.H. Taubes, Self-dual connections on 4-manifolds with indefinite intersection matrix, J.Diff.Geom. 19 (1984) 517–560.

    Article  MathSciNet  MATH  Google Scholar 

  39. C.H. Taubes, unpublished manuscript.

    Google Scholar 

  40. M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994) N.2, 317–353.

    Article  MathSciNet  MATH  Google Scholar 

  41. V.G. Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk, 41 (1986) N.1 (247) 97–147.

    MathSciNet  MATH  Google Scholar 

  42. K. Walker, An extension of Casson’s invariant Princeton Univ. Press, 1992.

    MATH  Google Scholar 

  43. B.L. Wang, Seiberg-Witten Floer theory for homology 3-spheres, preprint dg-ga/9602003.

    Google Scholar 

  44. G. Wang, R. Ye, Equivariant and Bott-type Seiberg-Witten Floer homology: Part I and Part II, preprint.

    Google Scholar 

  45. R.G. Wang, On Seiberg-Witten Floer invariants and the generalized Thom problem, preprint.

    Google Scholar 

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© 1999 Hindustan Book Agency

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Marcolli, M. (1999). Seiberg—Witten on three-manifolds. In: Seiberg-Witten Gauge Theory. Texts and Readings in Mathematics, vol 17. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-00-2_3

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