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Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory
 
Kenji Fukaya Stony Brook University, New York and Institute for Basic Sciences, Pohang, Korea
Yong-Geun Oh Institute for Basic Sciences, Pohang, Korea
Hiroshi Ohta Nagoya University, Japan
Kaoru Ono Kyoto University, Japan
Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory
Softcover ISBN:  978-1-4704-3625-4
Product Code:  MEMO/260/1254
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5325-1
Product Code:  MEMO/260/1254.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3625-4
eBook: ISBN:  978-1-4704-5325-1
Product Code:  MEMO/260/1254.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory
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Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory
Kenji Fukaya Stony Brook University, New York and Institute for Basic Sciences, Pohang, Korea
Yong-Geun Oh Institute for Basic Sciences, Pohang, Korea
Hiroshi Ohta Nagoya University, Japan
Kaoru Ono Kyoto University, Japan
Softcover ISBN:  978-1-4704-3625-4
Product Code:  MEMO/260/1254
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5325-1
Product Code:  MEMO/260/1254.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3625-4
eBook ISBN:  978-1-4704-5325-1
Product Code:  MEMO/260/1254.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2602019; 266 pp
    MSC: Primary 53; Secondary 14; 20

    In this paper the authors first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entov-Polterovich theory of spectral symplectic quasi-states and quasi-morphisms by incorporating bulk deformations, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher in a slightly less general context. Then the authors explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasi-morphisms and new Lagrangian intersection results on toric and non-toric manifolds.

    The most novel part of this paper is its use of open-closed Gromov-Witten-Floer theory and its variant involving closed orbits of periodic Hamiltonian system to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasi-morphism to the Lagrangian Floer theory (with bulk deformation).

    The authors use this open-closed Gromov-Witten-Floer theory to produce new examples. Using the calculation of Lagrangian Floer cohomology with bulk, they produce examples of compact symplectic manifolds \((M,\omega)\) which admits uncountably many independent quasi-morphisms \(\widetilde{{\rm Ham}}(M,\omega) \to {\mathbb{R}}\). They also obtain a new intersection result for the Lagrangian submanifold in \(S^2 \times S^2\).

  • Table of Contents
     
     
    • Chapters
    • Preface
    • 1. Introduction
    • 1. Review of spectral invariants
    • 2. Hamiltonian Floer-Novikov complex
    • 3. Floer boundary map
    • 4. Spectral invariants
    • 2. Bulk deformations of Hamiltonian Floer homology and spectral invariants
    • 5. Big quantum cohomology ring: Review
    • 6. Hamiltonian Floer homology with bulk deformations
    • 7. Spectral invariants with bulk deformation
    • 8. Proof of the spectrality axiom
    • 9. Proof of $C^0$-Hamiltonian continuity
    • 10. Proof of homotopy invariance
    • 11. Proof of the triangle inequality
    • 12. Proofs of other axioms
    • 3. Quasi-states and quasi-morphisms via spectral invariants with bulk
    • 13. Partial symplectic quasi-states
    • 14. Construction by spectral invariant with bulk
    • 15. Poincaré duality and spectral invariant
    • 16. Construction of quasi-morphisms via spectral invariant with bulk
    • 4. Spectral invariants and Lagrangian Floer theory
    • 17. Operator $\frak q$; review
    • 18. Criterion for heaviness of Lagrangian submanifolds
    • 19. Linear independence of quasi-morphisms.
    • 5. Applications
    • 20. Lagrangian Floer theory of toric fibers: review
    • 21. Spectral invariants and quasi-morphisms for toric manifolds
    • 22. Lagrangian tori in $k$-points blow up of $\mathbb {C}P^2$ ($k\ge 2$)
    • 23. Lagrangian tori in $S^2 \times S^2$
    • 24. Lagrangian tori in the cubic surface
    • 25. Detecting spectral invariant via Hochschild cohomology
    • 6. Appendix
    • 26. $\mathcal {P}_{(H_\chi ,J_\chi ),\ast }^{\frak b}$ is an isomorphism
    • 27. Independence on the de Rham representative of $\frak b$
    • 28. Proof of Proposition 20.7
    • 29. Seidel homomorphism with bulk
    • 30. Spectral invariants and Seidel homomorphism
    • 7. Kuranishi structure and its CF-perturbation: summary
    • 31. Kuranishi structure and good coordinate system
    • 32. Strongly smooth map and fiber product
    • 33. CF perturbation and integration along the fiber
    • 34. Stokes’ theorem
    • 35. Composition formula
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2602019; 266 pp
MSC: Primary 53; Secondary 14; 20

In this paper the authors first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entov-Polterovich theory of spectral symplectic quasi-states and quasi-morphisms by incorporating bulk deformations, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher in a slightly less general context. Then the authors explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasi-morphisms and new Lagrangian intersection results on toric and non-toric manifolds.

The most novel part of this paper is its use of open-closed Gromov-Witten-Floer theory and its variant involving closed orbits of periodic Hamiltonian system to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasi-morphism to the Lagrangian Floer theory (with bulk deformation).

The authors use this open-closed Gromov-Witten-Floer theory to produce new examples. Using the calculation of Lagrangian Floer cohomology with bulk, they produce examples of compact symplectic manifolds \((M,\omega)\) which admits uncountably many independent quasi-morphisms \(\widetilde{{\rm Ham}}(M,\omega) \to {\mathbb{R}}\). They also obtain a new intersection result for the Lagrangian submanifold in \(S^2 \times S^2\).

  • Chapters
  • Preface
  • 1. Introduction
  • 1. Review of spectral invariants
  • 2. Hamiltonian Floer-Novikov complex
  • 3. Floer boundary map
  • 4. Spectral invariants
  • 2. Bulk deformations of Hamiltonian Floer homology and spectral invariants
  • 5. Big quantum cohomology ring: Review
  • 6. Hamiltonian Floer homology with bulk deformations
  • 7. Spectral invariants with bulk deformation
  • 8. Proof of the spectrality axiom
  • 9. Proof of $C^0$-Hamiltonian continuity
  • 10. Proof of homotopy invariance
  • 11. Proof of the triangle inequality
  • 12. Proofs of other axioms
  • 3. Quasi-states and quasi-morphisms via spectral invariants with bulk
  • 13. Partial symplectic quasi-states
  • 14. Construction by spectral invariant with bulk
  • 15. Poincaré duality and spectral invariant
  • 16. Construction of quasi-morphisms via spectral invariant with bulk
  • 4. Spectral invariants and Lagrangian Floer theory
  • 17. Operator $\frak q$; review
  • 18. Criterion for heaviness of Lagrangian submanifolds
  • 19. Linear independence of quasi-morphisms.
  • 5. Applications
  • 20. Lagrangian Floer theory of toric fibers: review
  • 21. Spectral invariants and quasi-morphisms for toric manifolds
  • 22. Lagrangian tori in $k$-points blow up of $\mathbb {C}P^2$ ($k\ge 2$)
  • 23. Lagrangian tori in $S^2 \times S^2$
  • 24. Lagrangian tori in the cubic surface
  • 25. Detecting spectral invariant via Hochschild cohomology
  • 6. Appendix
  • 26. $\mathcal {P}_{(H_\chi ,J_\chi ),\ast }^{\frak b}$ is an isomorphism
  • 27. Independence on the de Rham representative of $\frak b$
  • 28. Proof of Proposition 20.7
  • 29. Seidel homomorphism with bulk
  • 30. Spectral invariants and Seidel homomorphism
  • 7. Kuranishi structure and its CF-perturbation: summary
  • 31. Kuranishi structure and good coordinate system
  • 32. Strongly smooth map and fiber product
  • 33. CF perturbation and integration along the fiber
  • 34. Stokes’ theorem
  • 35. Composition formula
Review Copy – for publishers of book reviews
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