Acta Mathematica Sinica

, Volume 15, Issue 1, pp 53–80

On the equivalence of multiplicative structures in floer homology and quantum homology

  • Gang Liu
  • Gang Tian


In this paper, we will prove that Floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring. We will also prove that GW-invariants in Floer homology and quantum homology are equivalent.


Floer homology Quantum homology GW-invariants J-holomorphic curves Moduli space 

1991MR Subject Classification

58F05 53C15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G Liu, G Tian. Weinstein conjecture and GW-invariants. preprint, 1997Google Scholar
  2. [2]
    Y Ruan, G Tian. Bott-type symplectic Floer cohomology and its multiplication structures. preprint, 1994Google Scholar
  3. [3]
    S Piunikhin, D Salamon, M schwarz. Symplectic Floer-Donoldson theory and quantum cohomology. preprint, 1994Google Scholar
  4. [4]
    G Liu. The equivalence of quantum cohomology and Floer cohomology. preprint, 1995Google Scholar
  5. [5]
    G Liu, G Tian. Floer homology and Arnold conjecture. J D G, to appearGoogle Scholar
  6. [6]
    G Liu, G Tian. Bott-type Floer homology. in preparationGoogle Scholar
  7. [7]
    Fukaya, Ono. Arnold conjecture and Gromov-Witten invariants. preprint, 1996Google Scholar
  8. [8]
    Y Long. Maslov-type index, degenerate critical points, and asymptotically linaer Hamiltonian systems. Science in China, Series A, 1990, 33: 1409–1419 (English edition).MATHGoogle Scholar
  9. [9]
    Y Long. A Maslov-type index theory fro symplectic paths. Top Meth Nonl Anal, 1997, 10: 47–78MATHGoogle Scholar
  10. [10]
    D Salamon, E Zehnder. Morse theory for periodic solutions of Hamiltonian equations and Maslov index. Comm Pure Appl Math, 1992, XLV(10): 1303–1360CrossRefGoogle Scholar
  11. [11]
    J Li, G Tian. Virtual moduli cycles and GW-invariants of general symplectic manifolds. Proceedings of 1st IP conference at UC, Irvine, 1996,Google Scholar
  12. [12]
    A Floer. Symplectic fixed points and holomorphic spheres. Comm Math Phys, 1989, 120: 575–611MATHCrossRefGoogle Scholar
  13. [13]
    M Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent Math, 1985, 82: 307–347MATHCrossRefGoogle Scholar
  14. [14]
    A Floer. The unregularized gradient flow of the symplectic action. Comm Pure Appl Math, 1988, 41: 775–813MATHCrossRefGoogle Scholar
  15. [15]
    G Liu. Associativity of quantum multiplication. Comm Math Phy, 1998Google Scholar
  16. [16]
    Y Ruan, G Tian. A mathematical theory of quantum cohomology. J Diff Geom, 1995, 42(2):Google Scholar
  17. [17]
    A Floer, H Hofer. Coherent orientations for periodic orbit problems in symplectic geometry. Math Zeit, 1994, 212: 13–38CrossRefGoogle Scholar
  18. [18]
    H Hofer, D A Salamon. Floer homology and Novikov rings. The Floer Memorial Volume, Progress in Math, 133: 483–524.Google Scholar
  19. [19]
    J Li, G Tian. Virtual moduli cycles and GW-invariants of algebraic varieties. J Amer Math Soc, 1998, 11(1): 119–174MATHCrossRefGoogle Scholar
  20. [20]
    D McDuff, D A Salamon. J-holomorphic curves and Quantum Cohomology. University Lecture Series, Amer Math Soc, Providence 1994Google Scholar
  21. [21]
    T H Parker, J G Wolfson. Pseudoholomorphic maps and bubble trees. Journ Geom Anal, 1993, 3: 63–98MATHGoogle Scholar
  22. [22]
    E Witten. Supersymmetry and Morse Theory. J Diff Geom, 1982, 17: 661–692MATHGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Gang Liu
    • 1
  • Gang Tian
    • 2
  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Department of MathematicsMITCambridgeUSA

Personalised recommendations