Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory.

Cover -- Title page -- Preface -- Chapter 1. Introduction -- 1.1. Introduction -- 1.2. Notations and Conventions -- 1.3. Difference between Entov-Polterovich's convention and ours -- Part 1 . Review of spectral invariants -- Chapter 2. Hamiltonian Floer-Novikov complex -- Chapter 3. Floer boundary map -- Chapter 4. Spectral invariants -- Part 2 . Bulk deformations of Hamiltonian Floer homology and spectral invariants -- Chapter 5. Big quantum cohomology ring: Review -- Chapter 6. Hamiltonian Floer homology with bulk deformations -- Chapter 7. Spectral invariants with bulk deformation -- Chapter 8. Proof of the spectrality axiom -- 8.1. Usher's spectrality lemma -- 8.2. Proof of nondegenerate spectrality -- Chapter 9. Proof of ⁰-Hamiltonian continuity -- Chapter 10. Proof of homotopy invariance -- Chapter 11. Proof of the triangle inequality -- 11.1. Pants products -- 11.2. Multiplicative property of Piunikhin isomorphism -- 11.3. Wrap-up of the proof of triangle inequality -- Chapter 12. Proofs of other axioms -- Part 3 . Quasi-states and quasi-morphisms via spectral invariants with bulk -- Chapter 13. Partial symplectic quasi-states -- Chapter 14. Construction by spectral invariant with bulk -- 14.1. Existence of the limit -- 14.2. partial quasi-morphism property of ₑ^{\frak } -- 14.3. Partial symplectic quasi-state property of ^{\frak }ₑ -- Chapter 15. Poincaré duality and spectral invariant -- 15.1. Statement of the result -- 15.2. Algebraic preliminary -- 15.3. Duality between Floer homologies -- 15.4. Duality and Piunikhin isomorphism -- 15.5. Proof of Theorem 1.1 -- Chapter 16. Construction of quasi-morphisms via spectral invariant with bulk -- Part 4 . Spectral invariants and Lagrangian Floer theory -- Chapter 17. Operator \frak -- review -- Chapter 18. Criterion for heaviness of Lagrangian submanifolds -- 18.1. Statement of the results.

18.2. Floer homologies of periodic Hamiltonians and of Lagrangian submanifolds -- 18.3. Filtration and the map \frak _{( , )}^{ ,\frak } -- 18.4. Identity \frak _{( , )}^{ ,\frak ,∗}∘\CP_{( ᵪ, ),∗}^{\frak }= _{ , }* -- 18.5. Heaviness of -- Chapter 19. Linear independence of quasi-morphisms. -- Part 5 . Applications -- Chapter 20. Lagrangian Floer theory of toric fibers: review -- 20.1. Toric manifolds: review -- 20.2. Review of Floer cohomology of toric fiber -- 20.3. Relationship with the Floer cohomology in Chapter 17 -- 20.4. Properties of Floer cohomology _{ }(( , ) -- Λ): review -- Chapter 21. Spectral invariants and quasi-morphisms for toric manifolds -- 21.1. ₑ^{\frak }-heaviness of the Lagrangian fibers in toric manifolds -- 21.2. Calculation of the leading order term of the potential function in the toric case: review -- 21.3. Existence of Calabi quasi-morphism on toric manifolds -- 21.4. Defect estimate of a quasi-morphism ₑ^{\frak } -- Chapter 22. Lagrangian tori in -points blow up of \C ² ( ≥2) -- Chapter 23. Lagrangian tori in ²× ² -- 23.1. Review of the construction from [FOOO6] -- 23.2. Superheaviness of ( ) -- 23.3. Critical values and eigenvalues of ₁( ) -- Chapter 24. Lagrangian tori in the cubic surface -- Chapter 25. Detecting spectral invariant via Hochschild cohomology -- 25.1. Hochschild cohomology of filtered _{∞} algebra: review -- 25.2. From quantum cohomology to Hochschild cohomology -- 25.3. Proof of Theorem 25.1 -- 25.4. A remark -- Part 6 . Appendix -- Chapter 26. \CP_{( ᵪ, ᵪ),∗}^{\frak } is an isomorphism -- Chapter 27. Independence on the de Rham representative of \frak -- Chapter 28. Proof of Proposition 20.7 -- 28.1. Pseudo-isotopy of filtered _{∞} algebra -- 28.2. Difference between \frak ^{ } and \frak -- 28.3. Smoothing ⁿ-invariant chains -- 28.4. Wrap-up of the proof of Proposition 3.1.

28.5. Proof of Lemma 3.3 -- Chapter 29. Seidel homomorphism with bulk -- 29.1. Seidel homomorphism with bulk -- 29.2. Proof of Theorem 29.13 -- 29.3. Proof of Theorem 29.9 -- Chapter 30. Spectral invariants and Seidel homomorphism -- 30.1. Valuations and spectral invariants -- 30.2. The toric case -- Part 7 . Kuranishi structure and its CF-perturbation: summary -- Chapter 31. Kuranishi structure and good coordinate system -- 31.1. Orbifold -- 31.2. Kuranishi structure -- Chapter 32. Strongly smooth map and fiber product -- 32.1. Strongly smooth map -- 32.2. Fiber product -- Chapter 33. CF perturbation and integration along the fiber -- 33.1. Differential form on the space with Kuranishi structure -- 33.2. CF-perturbation -- 33.3. Integration along the fiber -- Chapter 34. Stokes' theorem -- 34.1. Normalized boundary -- 34.2. Statement of Stokes' theorem -- Chapter 35. Composition formula -- 35.1. Smooth correspondence and its perturbation -- 35.2. Composition of smooth correspondences -- 35.3. Statement of Composition formula -- Bibliography -- Index -- Back Cover.

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.

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