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

Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru

doi:10.1090/memo/1254

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

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Kenji Fukaya, Simons Center for Geometry and Physics, State University of New York, Stony Brook, New York 11794-3636 — and — Center for Geometry and Physics, Institute for Basic Sciences (IBS), 77 Cheongam-ro, Nam-gu, Pohang, Republic of Korea, Yong-Geun Oh, Center for Geometry and Physics, Institute for Basic Sciences (IBS), 77 Cheongam-ro, Nam-gu, Pohang, Republic of Korea — and — Department of Mathematics, POSTECH, Pohang, Republic of Korea, Hiroshi Ohta, Graduate School of Mathematics, Nagoya University, Nagoya, Japan and Kaoru Ono, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 260, Number 1254
ISBNs: 978-1-4704-3625-4 (print); 978-1-4704-5325-1 (online)
DOI: https://doi.org/10.1090/memo/1254
Published electronically: July 17, 2019
Keywords: Floer homology, Lagrangian submanifolds, Hamiltonian dynamics, bulk deformations, spectral invariants, partial symplectic quasi-states, quasi-morphisms, quantum cohomology, toric manifold, open-closed Gromov-Witten theory
MSC: Primary 53D40, 53D12, 53D45; Secondary 53D20, 14M25, 20F65

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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. Quasi-states and quasi-morphisms via spectral invariants with bulk

13. Partial symplectic quasi-states
14. Construction by spectral invariant with bulk
15. Poincaré duality and spectral invariant
16. Construction of quasi-morphisms 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 quasi-morphisms.

5. Applications

20. Lagrangian Floer theory of toric fibers: review
21. Spectral invariants and quasi-morphisms 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 CF-perturbation: 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

Abstract

In this paper we 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 (2011) in a slightly less general context. Then we 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 to use open-closed Gromov-Witten-Floer theory (operator $\frak q$ in Fukaya, et al. (2009) 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).

We use this open-closed Gromov-Witten-Floer theory to produce new examples. Especially using the calculation of Lagrangian Floer cohomology with bulk deformation in Fukaya, et al. (2010, 2011, 2016), we produce examples of compact symplectic manifolds $(M,\omega )$ which admits uncountably many independent quasi-morphisms $\widetilde {\operatorname {Ham}}(M,\omega ) \to \mathbb {R}$. We also obtain a new intersection result for the Lagrangian submanifold in $S^2 \times S^2$ discovered in Fukaya, et al. (2012).

Many of these applications were announced in Fukaya, et al. (2010, 2011, 2012).

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

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