Communications in Mathematical Physics

, Volume 118, Issue 2, pp 215–240 | Cite as

An instanton-invariant for 3-manifolds

  • Andreas Floer
Article

Abstract

To an oriented closed 3-dimensional manifoldM withH1(M, ℤ)=0, we assign a ℤ8-graded homology groupI*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×ℝ.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ad]
    Adams, R.A.: Sobolev spaces. New York: Academic Press 1975Google Scholar
  2. [Ar]
    Arondzajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of the second order. J. Math. Pures Appl.36 (9), 235–249 (1957)Google Scholar
  3. [A1]
    Atiyah, M.F.: Instantons in two and four dimensions. Commun. Math. Phys.93, 437–451 (1984)Google Scholar
  4. [A2]
    Atiyah, M.F.: New invariants for 3- and 4-dimensional manifolds. Preprint 1987Google Scholar
  5. [ADHM]
    Atiyah, M.F., Drinfield, V., Hitchin, N., Manin, Y.I.: Construction of instantons. Phys. Lett. A65, 185 (1978)Google Scholar
  6. [AS1]
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators I. Ann. Math.87, 484–530 (1968); III. Ibid.87, 546–604 (1968)Google Scholar
  7. [AS2]
    Atiyah, M.F.: Index theory of skew adjoint Fredholm operators. Publ. Math. IHES37, 305–325 (1969)Google Scholar
  8. [B]
    Braam, P.: Monopoles on three-manifolds, preprint, Oxford, 1987Google Scholar
  9. [D1]
    Donaldson, S.K.: An application of gauge theory to the topology of 4-manifolds. J. Differ. Geom.18, 269–316 (1983)Google Scholar
  10. [D2]
    Donaldson, S.K.: Instantons and geometric invariant theoryGoogle Scholar
  11. [D3]
    Donaldson, S.K.: The orientation of Yang-Mills moduli-spaces and four manifold topology. J. Differ. Geom.26, 397–428 (1987)Google Scholar
  12. [D4]
    Donaldson, S.K.: Polynomial invariants for smooth 4-manifolds. Preprint, Oxford, 1987Google Scholar
  13. [FS]
    Fintushel, R., Stern, R.J.: Pseudofree orbifolds. Ann. Math.122, 335–346 (1985)Google Scholar
  14. [F1]
    Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. (to appear)Google Scholar
  15. [F2]
    Floer, A.: The unregularized gradient flow for the symplectic action. Commun. Pure Appl. Math. (to appear)Google Scholar
  16. [F3]
    Floer, A.: Symplectic fixed points and holomorphic spheres. Preprint, CIMS, 1988Google Scholar
  17. [FU]
    Freed, D., Uhlenbeck, K.K.: Instantons and four-manifolds. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  18. [G]
    Goldman, W.: The symplectic nature of the fundamental groups of surfaces. Adv. Math.54, 200–225 (1984)Google Scholar
  19. [He]
    Hempel, J.: Three-manifolds. Ann. of Math. Studies 86. Princeton, NJ: Princeton, University Press 1967Google Scholar
  20. [Hö]
    Hörmander, L.: The analysis of linear differential operators. III. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  21. [Hu]
    Husemoller, D.: Fibre bundles. Springer Graduate Texts in Math., Vol. 20. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  22. [K]
    Kondrat'ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc.16 (1967)Google Scholar
  23. [LM]
    Lockhard, R.B., McOwen, R.C.: Elliptic operators on noncompact manifolds. Ann. Sci. Norm. Sup. PisaIV-12, 409–446 (1985)Google Scholar
  24. [Mc]
    Matic, G.: SO(3) connections and rational homology cobordism. Preprint, MIT, 1987Google Scholar
  25. [MP]
    Maz'ja, V.G., Plamenevski, B.A.: Estimates onL p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary problems in domains with singular points on the boundary. Math. Nachr.81, 25–82 (1978) [English Transl. In: AMS Translations Ser. 2,123, 1–56 (1984)Google Scholar
  26. [Mr]
    Milnor, J.: Lectures on theh-cobordism theorem. Math. Notes. Princeton, NJ: Princeton University Press 1965Google Scholar
  27. [N]
    Novikov, S.P.: Multivalued functions and functionals, an analogue of Morse theory. Sov. Math. Dokl.24, 222–226 (1981)Google Scholar
  28. [P]
    Palais, R.S.: Foundations of global analysis. New York: Benjamin 1968Google Scholar
  29. [Q]
    Quinn, F.: Transversal approximation on Banach manifolds. In: Proc. Symp. Pure Math. 15. Providence, RI: AMS 1970Google Scholar
  30. [RS]
    Ray, D.B., Singer, I.M.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145–201 (1971)Google Scholar
  31. [S1]
    Smale, S.: Morse inequalities for dynamical systems. Bull. AMS66, 43–49 (1960)Google Scholar
  32. [S2]
    Smale, S.: On gradient dynamical systems. Ann. Math.74, 199–206 (1961)Google Scholar
  33. [S3]
    Smale, S.: An infinite dimensional version of Sard's theorem. Am. J. Math.87, 213–221 (1973)Google Scholar
  34. [Sp]
    Spanier, E.: Algebraic topology. New York: McGraw-Hill 1966Google Scholar
  35. [T1]
    Taubes, C.H.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differ. Geom.17, 139–170 (1982)Google Scholar
  36. [T2]
    Taubes, C.H.: Gauge theory on asymptotically periodic 4-manifolds. J. Differ. Geom.25, 363–430 (1987)Google Scholar
  37. [T3]
    Taubes, C.H.: Private communicationGoogle Scholar
  38. [T4]
    Taubes, C.H.: A framework for Morse theory for the Yang-Mills functional. Preprint, Harvard, 1986Google Scholar
  39. [T5]
    Taubes, C.H.: Preprint, 1987Google Scholar
  40. [U1]
    Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11–29 (1982)Google Scholar
  41. [U2]
    Uhlenbeck, K.K.: Connections withL p-bounds on curvature. Commun. Math. Phys.83, 31–42 (1982)Google Scholar
  42. [W1]
    Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom.17, 661–692 (1982)Google Scholar
  43. [W2]
    Witten, E.: Topological quantum field theory. Commun. Math. Phys.117, 353–386 (1988)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Andreas Floer
    • 1
  1. 1.Courant InstituteNew YorkUSA

Personalised recommendations