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Harmonic Spinors and Topology

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New Developments in Differential Geometry, Budapest 1996

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Abstract

We discuss relations between the dimension of the solution space of the Dirac equation and the topology of the underlying manifold. It is shown that in certain dimensions existence of metrics with harmonic spinors is not topologically obstructed. In this respect the Dirac operator behaves very differently from the Laplace-Beltrami operator.

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References

  1. M.P. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian Geometry I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69

    Article  MathSciNet  MATH  Google Scholar 

  2. M.F. Atiyah, I.M. Singer, The index of elliptic operators: III, Ann. Math. 87 (1968), 546–604

    Article  MathSciNet  MATH  Google Scholar 

  3. M.F. Atiyah, I.M. Singer, The index of elliptic operators: IV, Ann. Math. 93 (1971), 119–138

    Article  MathSciNet  MATH  Google Scholar 

  4. M.F. Atiyah, I.M. Singer, The index of elliptic operators: V, Ann. Math. 93 (1971), 139–149

    Article  MathSciNet  Google Scholar 

  5. C. Bär, Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39–46

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Bär, Metrics with harmonic spinors, GAFA 6 (1996), 899–942

    Article  MATH  Google Scholar 

  7. C. Bär, Harmonic spinors for twisted Dirac operators, Math. Ann. 309 (1997), 225–246

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Bär, P. Schmutz, Harmonic spinors on Riemann surfaces, Ann. Glob. Anal. Geom. 10 (1992), 263–273

    Article  MATH  Google Scholar 

  9. N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators, Springer 1991

    Google Scholar 

  10. S.K. Donaldson, The Seiberg-Witten equations and 4-manifold theory, Bull. Amer. Math. Soc. 3 (1996), 45–70

    Article  MathSciNet  Google Scholar 

  11. T. Friedrich, Zur Abhängigkeit des Dirac-Operators von der Spin-Struktur, Collect. Math. 48 (1984), 57–62

    MathSciNet  MATH  Google Scholar 

  12. M. Gromov, H.-B. Lawson, Spin and scalar curvature in the presence of a fundamental group I, Ann. Math. 111 (1980), 209–230

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Gromov, H.-B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. Math. 111 (1980), 423–434

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Gromov, H.-B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. I.H.E.S. 58 (1983), 295–408

    Google Scholar 

  15. P. Hirzebruch, H. Hopf, Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Ann. 136 (1958), 156–172

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Katz, A proof via the Seiberg-Witten moduli space of Donaldson’s theorem on smooth 4-manifolds with definite intersection form, Preprint

    Google Scholar 

  18. J.L. Kazdan, F.W. Warner, Existence and conformai deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. Math. 101 (1975), 317–331

    Article  MathSciNet  MATH  Google Scholar 

  19. J.L. Kazdan, F.W. Warner, Scalar curvature and conformai deformation of Riemannian structure, J. Diff. Geom. 10 (1975), 113–134

    MathSciNet  MATH  Google Scholar 

  20. D. Kotschick, Non-trivial harmonic spinors on certain algebraic surfaces, in ‘Einstein metrics and Yang-Mills connections’, ed. T. Mabuchi and S. Mukai, Marcel Dekker, New York, Basel, Hong Kong 1993

    Google Scholar 

  21. D. Kotschick, Non-trivial harmonic spinors on generic algebraic surfaces, Proc. Am. Math. Soc. 124 (1996), 2315–2318

    Article  MathSciNet  MATH  Google Scholar 

  22. P.B. Kronheimer, T.S. Mrowka, The genus of embedded surfaces in the protective plane, Math. Res. Lett. 1 (1994), 797–808

    MathSciNet  MATH  Google Scholar 

  23. H.-B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, Princeton 1989

    MATH  Google Scholar 

  24. A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 7–9

    MathSciNet  MATH  Google Scholar 

  25. S. Maier, Generic Metrics and Connections on Spin-and Spin c-Manifolds, Preprint 1996

    Google Scholar 

  26. J.W. Milnor, Remarks concerning spin manifolds, in: S. Cairns (Ed.), Differential and Combinatorial Topology, Princeton 1965, 55–62

    Google Scholar 

  27. J.W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes 44, Princeton University Press, Princeton 1996

    MATH  Google Scholar 

  28. J. Rosenberg, S. Stolz, Manifolds of positive scalar curvature, In: G.E. Carlsson et al. (Ed), Algebraic Topology and its Applications, Springer Math. Sci. Res. Inst. Publ. 27 (1994), 241–267

    Google Scholar 

  29. T. Schick, A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Preprint 1997

    Google Scholar 

  30. R. Schoen, S.T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159–183

    Article  MathSciNet  MATH  Google Scholar 

  31. N. Seiberg, E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nuclear Phys. B 431 (1994), 484–550

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Stolz, Simply connected manifolds of positive scalar curvature, Bull. Amer. Math. Soc. 23 (1990), 427–432

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. Math. 136 (1992), 511–540

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Stolz, Positive scalar curvature metrics — existence and classification questions, Proc. Intern. Congr. of Mathematicians Zürich Vol. 1 (1994), 625–636

    MathSciNet  Google Scholar 

  35. M. Wang, Parallel spinors and parallel forms, Ann. Glob. Anal. Geom. 7 (1989), 59–68

    Article  MATH  Google Scholar 

  36. M. Wang, On non-simply connected manifolds with non-trivial parallel spinors, Ann. Glob. Anal. Geom. 13 (1995), 31–42

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104, Freiburg, Germany

    Christian Bär

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  1. Christian Bär
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Editors and Affiliations

  1. Department of Geometry, Loránd Eötvös University, Budapest, Hungary

    J. Szenthe

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© 1999 Springer Science+Business Media Dordrecht

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Bär, C. (1999). Harmonic Spinors and Topology. In: Szenthe, J. (eds) New Developments in Differential Geometry, Budapest 1996. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5276-1_3

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  • DOI: https://doi.org/10.1007/978-94-011-5276-1_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6220-6

  • Online ISBN: 978-94-011-5276-1

  • eBook Packages: Springer Book Archive

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