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OverviewIn 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 $S^2 \times S^2$. Full Product DetailsAuthor: Kenji Fukaya , Yong-Geun Oh , Hiroshi Ohta , Kaoru OnoPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.396kg ISBN: 9781470436254ISBN 10: 1470436256 Pages: 262 Publication Date: 30 October 2019 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction 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.ReviewsAuthor InformationKenji 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. Tab Content 6Author Website:Countries AvailableAll regions |
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