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.

• IntroductionPart 1. Review of spectral invariants: Hamiltonian Floer-Novikov complexFloer boundary mapSpectral invariantsPart 2. Bulk deformations of Hamiltonian Floer homology and spectral invariants: Big quantum cohomology ring: reviewHamiltonian Floer homology with bulk deformationsSpectral invariants with bulk deformationProof of the spectrality axiomProof of $C^0$-Hamiltonian continuityProof of homotopy invarianceProof of the triangle inequalityProofs of other axiomsPart 3. Quasi-states and quasi-morphisms via spectral invariants with bulk: Partial symplectic quasi-statesConstruction by spectral invariant with bulkPoincare duality and spectral invariantConstruction of quasi-morphisms via spectral invariant with bulkPart 4. Spectral invariants and Lagrangian Floer theory: Operator $\mathfrak q$reviewCriterion for heaviness of Lagrangian submanifoldsLinear independence of quasi-morphismsPart 5. Applications: Lagrangian Floer theory of toric fibers: reviewSpectral invariants and quasi-morphisms for toric manifoldsLagrangian 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 surfaceDetecting spectral invariant via Hochschild cohomologyPart 6. Appendix: $\mathcal {P}_{(H_\chi ,J_\chi ),\ast }^{\mathfrak b}$ is an isomorphismIndependence on the de Rham representative of $\mathfrak b$Proof of Proposition 20.7Seidel homomorphism with bulkSpectral invariants and Seidel homomorphismPart 7. Kuranishi structure and its CF-perturbation: summary: Kuranishi structure and good coordinate systemStrongly smooth map and fiber productCF perturbation and integration along the fiberStokes' theoremComposition formulaBibliographyIndex.