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Gauge Theory and Four-Manifold Topology

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First European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 3))

Abstract

This article is a survey of recent developments in the area described in the title. We will concentrate on the developments that have taken place since 1986, after the earlier survey [10] (which corresponds roughly to the material covered in the book [14]), and this article might be viewed as a continuation of [10]. Another useful reference which gives an overview of the field is [23].

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Reference

  1. M. F. Atiyah, New invariants of 3 and 4-dimensional manifolds, in: The Mathematical Hertitage of Hermann Weyl, Proc. Svmpos. Pure Maths, 48, 1988.

    Google Scholar 

  2. M.F. Atiyah, Topological Quantum Field Theories, PublMath. IHES 68 (1988), 175–186.

    MathSciNet  MATH  Google Scholar 

  3. W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Springer, Heidelberg, 1984.

    MATH  Google Scholar 

  4. S. Bauer, Some nonreduced moduli of bundles and Donaldson invariants for Dolgachev surfaces, Jour, für die reine angew. Math. 424 (1992), 149–180.

    Article  MATH  Google Scholar 

  5. S. Bauer, Diffeomorphism types of elliptic surfaces with pg = 1, Submitted to Jour, für die reine angew. math..

    Google Scholar 

  6. S. Bauer and C. Okonek, The algebraic geometry of representation spaces associated to Seifert-fibred homology 3-spheres, Math. Annalen 286 (1990), 45–76.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. U. Boden, Representations of orbifold groups and parabolic bundles, Commun. Math. Helveticii 66 (1991), 389–447.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Brussee, Stable bundles on blown-up surfaces, Math. Z. 205 (1990), 551–565.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. K. Donaldson, Irrationality and the h-cobordism conjecture, Jour. Differential Geometry 26 (1987), 141–168.

    MathSciNet  MATH  Google Scholar 

  10. S. K. Donaldson, The geometry of 4-manifolds, Proc. Int. Congress Math., Berkeley, 1 (1986), 43–61.

    Google Scholar 

  11. S. K. Donaldson, Polynomial invariants for smooth 4-manifolds, Topology 29 (1990), 257–315.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. K. Donaldson, Complex curves and surgery, Math. Publ. IHES 68 1988, 91–97.

    MathSciNet  MATH  Google Scholar 

  13. S. K. Donaldson, Gluing techniques in the cohomology of moduli spaces, To appear in Vol. dedicated to J. Milnor (Publish and Perish).

    Google Scholar 

  14. S. K. Donaldson and P. B. Kronheimer, The geometry of 4-manifolds, Oxford University Press, Oxford, 1990.

    Google Scholar 

  15. S. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves, Warwick University, preprint, 1992.

    Google Scholar 

  16. W. Ebeling, An example of two homeomorphic, nondiffeomorphic complete intersection surfaces, Inventiones Math. 99 (1990), 651–654.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Fintushel and R. Stern, Instanton homology of Seifert fibered homology three-spheres, Proc. Loud. Math. Soc. 61 (1990), 109–137.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Fintushel and R. Stern, 2-torsion instanton invariants, preprint, 1991.

    Google Scholar 

  19. A. Floer, An instanton invariant for 3-manifolds, Commun. Math. Phys. 118 (1988), 215–240.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Floer, Symplectic fixed points and holomorphic spheres, Commun. Math. Phys. 120 (1989), 575–611.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Friedman, Rank two vector bundles over regular elliptic surfaces Inventiones Math. 96 (1989), 283–332.

    Article  MATH  Google Scholar 

  22. R. Friedman and J.W. Morgan, On the diffeomorphism types of certain algebraic surfaces, I, Jour. Differential Geometry 27 (1988), 297–369.

    MathSciNet  MATH  Google Scholar 

  23. R. Friedman and J.W.Morgan, Algebraic surfaces and 4-manifolds: some conjectures and speculations, Bull. Amer. Math. Soc. 18 (1988), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Friedman and J. W. Morgan, Complex versus differentiate classification of algebraic surfaces, Columbia University, preprint, 1989.

    Google Scholar 

  25. R. Friedman and J. W. Morgan, Smooth 4-manifolds and complex surfaces, (to appear).

    Google Scholar 

  26. R. Friedman, B. Moishezon and J.W. Morgan, On the C invariance of the canonical class of certain algebraic surfaces, Bull. Amer. Math. Soc. 17 (1987), 283–286.

    Article  MathSciNet  MATH  Google Scholar 

  27. K. Fukaya, Floer homology of connected sums of homology 3-spheres, University of Tokyo, preprint, 1992.

    Google Scholar 

  28. M. Furuta and B. Steer, Seifert-fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Advances in Math. 96 (1992), 38–102.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. E. Gompf, Nuclei of elliptic surfaces, Topology, (to appear).

    Google Scholar 

  30. R. E. Gompf, Some new symplectic 4-manifolds, Max Planck Institut, preprint, 1992.

    Google Scholar 

  31. R. E. Gompf and T. S. Mrowka, A family of non-complex homotopy K3 surfaces, to appear in Annals of Math.

    Google Scholar 

  32. P.A. Kirk and E.P. Klassen, Representation spaces of Seifert fibered homology spheres, Topology 30 (1991), 77–95.

    Article  MathSciNet  MATH  Google Scholar 

  33. K. Kodaira, On homotopy K3 surfaces, Gollected Works, Vol III, Princeton University Press, Princeton.

    Google Scholar 

  34. D. Kotschick, On manifolds homeomorphic to \( C{P^2}\# 8{\overline {CP} ^2} \), Inventiones Math. 95 (1989), 591–600.

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Kotschick, On the geometry of certain 4-manifolds, Oxford D. Phil., Thesis (1989).

    Google Scholar 

  36. D. Kotschick, S0(3) invariants for 4-manifolds with \( b_2^ + \) = 1, Proc. Lond. Math. Soc. 63 (1991), 426–448.

    Article  MathSciNet  MATH  Google Scholar 

  37. D. Kotschick and J.W. Morgan, S0(3) invariants for 4-manifolds with \( b_2^ + \) = 1, I I, in preparation.

    Google Scholar 

  38. Kronheimer, Embedded surfaces in 4-manifolds, in: Proc. Int. Cong. Math (Kyoto) Vol I, 1990, 527–539.

    Google Scholar 

  39. P. B. Kronheimer, Instanton invariants and fiat connections on the Kummer surface, Duke Math. Jour. 64 (1991), 229–242.

    Article  MathSciNet  MATH  Google Scholar 

  40. P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces I,II, submitted to Topology.

    Google Scholar 

  41. P. Le Barz, Formules pour les trisecantes des surfaces algebriques, L’Enseignement Math. 33 (1987), 1–66.

    MATH  Google Scholar 

  42. W. Li and Z. Qin, On the Donaldson polynomial invariants of rational ruled surfaces, submitted to Jour. of Algebraic Geometry.

    Google Scholar 

  43. J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants of algebraic surfaces, to appear in Jour. Differential Geometry.

    Google Scholar 

  44. P. Lisca, New irreducible non-complex 4-manifolds, submitted to Jour. Differential Geometry.

    Google Scholar 

  45. K. C. Mong, Some invariants of differentiate 4-manifolds, to appear in Quarterly Jour. Math.

    Google Scholar 

  46. J. W. Morgan and T. S. Mrowka, A note on Donaldson’s polynomial invariants, to appear in Duke Math. Jour.

    Google Scholar 

  47. J. W. Morgan, Comparison of the Donaldson invariants of algebraic surfaces with their algebro-geometric analogues, to appear in Topology.

    Google Scholar 

  48. J.W. Morgan and K.G. O’Grady, The smooth classification of fake K3 and similar surfaces, Columbia University, preprint, 1992.

    Google Scholar 

  49. T. S. Mrowka, A local Mayer-Vietoris principle for the instanton moduli spaces over manifolds with cylindrical ends, Ph.D thesis, (Berkeley).

    Google Scholar 

  50. K.G. O’Grady, Donaldson’s polynomials for K3 surfaces, Jour. Differential Geometry 35 (1992), 415–428.

    MathSciNet  MATH  Google Scholar 

  51. K.G. O’Grady, Algebro-geometric analogues of Donaldson’s polynomials, to appear in Inventiones Math.

    Google Scholar 

  52. C. Okonek and A. Van de Ven, Stable bundles and differentiable structures on certain elliptic surfaces, Invent. Math. 86 1986, 357–370.

    Article  MathSciNet  MATH  Google Scholar 

  53. U. Persson and C. Peters, Homeomorphic non-diffeomorphic surfaces with small invariants, preprint.

    Google Scholar 

  54. —, Invariants of the smooth structure of an algebraic surface arising from the Dirac operator, V. Ya. Pidstragach and A. N. Tyurin, University of Warwick, preprint, 1992.

    Google Scholar 

  55. Z. Qin, Complex structures on certain differentiable 4-manifolds, to appear in Topology.

    Google Scholar 

  56. M. Salvetti, A lower bound for the number of differentiable structures on 4-manifolds, Boll, dell Unione Mat. Italiana 7 (1991), 33–40.

    MathSciNet  Google Scholar 

  57. C. H. Taubes, L2 moduli spaces on manifolds with cylindrical ends, Harvard University, preprint, 1991.

    Google Scholar 

  58. M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, Jour. Differential geometry 35 1992, 131–150.

    MathSciNet  MATH  Google Scholar 

  59. S. Wang, Moduli spaces over manifolds with involutions, to appear in Math. Annalen.

    Google Scholar 

  60. E. Witten, On quantum gauge theories in two dimensions, Commun. Math. Phys. 141 (1991), 153–209.

    Article  MathSciNet  MATH  Google Scholar 

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Donaldson, S.K. (1994). Gauge Theory and Four-Manifold Topology. In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics . Progress in Mathematics, vol 3. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9110-3_4

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  • DOI: https://doi.org/10.1007/978-3-0348-9110-3_4

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9911-6

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