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 $S^2 \times S^2$.
Introduction
Part
1. Review of spectral invariants: Hamiltonian Floer-Novikov complex
Floer boundary map
Spectral invariants
Part
2. Bulk deformations of Hamiltonian Floer homology and spectral
invariants: Big quantum cohomology ring: review
Hamiltonian Floer homology with bulk deformations
Spectral invariants with bulk deformation
Proof of the spectrality axiom
Proof of $C^0$-Hamiltonian continuity
Proof of homotopy invariance
Proof of the triangle inequality
Proofs of other axioms
Part
3. Quasi-states and quasi-morphisms via spectral invariants with bulk:
Partial symplectic quasi-states
Construction by spectral invariant with bulk
Poincare duality and spectral invariant
Construction of quasi-morphisms via spectral invariant with bulk
Part
4. Spectral invariants and Lagrangian Floer theory: Operator $\mathfrak
q$
review
Criterion for heaviness of Lagrangian submanifolds
Linear independence of quasi-morphisms
Part
5. Applications: Lagrangian Floer theory of toric fibers: review
Spectral invariants and quasi-morphisms for toric manifolds
Lagrangian tori in $k$-points blow up of $\mathbb {C}P^2$ ($k\ge 2$)
Lagrangian tori in $S^2 \times S^2$
Lagrangian tori in the cubic surface
Detecting spectral invariant via Hochschild cohomology
Part
6. Appendix: $\mathcal {P}_{(H_\chi ,J_\chi ),\ast }^{\mathfrak b}$ is
an isomorphism
Independence on the de Rham representative of $\mathfrak b$
Proof of Proposition 20.7
Seidel homomorphism with bulk
Spectral invariants and Seidel homomorphism
Part
7. Kuranishi structure and its CF-perturbation: summary: Kuranishi
structure and good coordinate system
Strongly smooth map and fiber product
CF perturbation and integration along the fiber
Stokes' theorem
Composition formula
Bibliography
Index.
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.
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