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

About this Title

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)
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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Table of Contents


  • Preface
  • 1. Introduction

1. Review of spectral invariants

  • 2. Hamiltonian Floer-Novikov complex
  • 3. Floer boundary map
  • 4. Spectral invariants

2. Bulk deformations of Hamiltonian Floer homology and spectral invariants

  • 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


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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