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Product Code:  MEMO/260/1254 
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Softcover ISBN:  9781470436254 
eBook ISBN:  9781470453251 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 260; 2019; 266 ppMSC: 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 EntovPolterovich theory of spectral symplectic quasistates and quasimorphisms 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 quasistates, quasimorphisms and new Lagrangian intersection results on toric and nontoric manifolds.
The most novel part of this paper is its use of openclosed GromovWittenFloer theory and its variant involving closed orbits of periodic Hamiltonian system to connect spectral invariants (with bulk deformation), symplectic quasistates, quasimorphism to the Lagrangian Floer theory (with bulk deformation).
The authors use this openclosed GromovWittenFloer 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 quasimorphisms \(\widetilde{{\rm Ham}}(M,\omega) \to {\mathbb{R}}\). 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 FloerNovikov 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. Quasistates and quasimorphisms via spectral invariants with bulk

13. Partial symplectic quasistates

14. Construction by spectral invariant with bulk

15. Poincaré duality and spectral invariant

16. Construction of quasimorphisms 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 quasimorphisms.

5. Applications

20. Lagrangian Floer theory of toric fibers: review

21. Spectral invariants and quasimorphisms 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 CFperturbation: 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


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In this paper the authors first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of EntovPolterovich theory of spectral symplectic quasistates and quasimorphisms 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 quasistates, quasimorphisms and new Lagrangian intersection results on toric and nontoric manifolds.
The most novel part of this paper is its use of openclosed GromovWittenFloer theory and its variant involving closed orbits of periodic Hamiltonian system to connect spectral invariants (with bulk deformation), symplectic quasistates, quasimorphism to the Lagrangian Floer theory (with bulk deformation).
The authors use this openclosed GromovWittenFloer 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 quasimorphisms \(\widetilde{{\rm Ham}}(M,\omega) \to {\mathbb{R}}\). 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 FloerNovikov 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. Quasistates and quasimorphisms via spectral invariants with bulk

13. Partial symplectic quasistates

14. Construction by spectral invariant with bulk

15. Poincaré duality and spectral invariant

16. Construction of quasimorphisms 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 quasimorphisms.

5. Applications

20. Lagrangian Floer theory of toric fibers: review

21. Spectral invariants and quasimorphisms 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 CFperturbation: 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