Spectral invariants with bulk, quasi-morphisms and Lagrangian floer theory / Kenji Fukaya [and three others].

Format:

Book

Author/Creator:

Fukaya, Kenji, author.

Oh, Yong-Geun, 1961- author.

Ohta, Hiroshi (Mathematician), author.

Ono, Kaoru (Mathematician), author.

Series:

Memoirs of the American Mathematical Society ; no. 1254.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1254

Language:

English

Subjects (All):

Symplectic geometry.

Lagrangian functions.

Floer homology.

Physical Description:

1 online resource (282 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2019]

Summary:

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.

Contents:

Cover

Title page

Preface

Chapter 1. Introduction

1.1. Introduction

1.2. Notations and Conventions

1.3. Difference between Entov-Polterovich's convention and ours

Part 1 . Review of spectral invariants

Chapter 2. Hamiltonian Floer-Novikov complex

Chapter 3. Floer boundary map

Chapter 4. Spectral invariants

Part 2 . Bulk deformations of Hamiltonian Floer homology and spectral invariants

Chapter 5. Big quantum cohomology ring: Review

Chapter 6. Hamiltonian Floer homology with bulk deformations

Chapter 7. Spectral invariants with bulk deformation

Chapter 8. Proof of the spectrality axiom

8.1. Usher's spectrality lemma

8.2. Proof of nondegenerate spectrality

Chapter 9. Proof of ⁰-Hamiltonian continuity

Chapter 10. Proof of homotopy invariance

Chapter 11. Proof of the triangle inequality

11.1. Pants products

11.2. Multiplicative property of Piunikhin isomorphism

11.3. Wrap-up of the proof of triangle inequality

Chapter 12. Proofs of other axioms

Part 3 . Quasi-states and quasi-morphisms via spectral invariants with bulk

Chapter 13. Partial symplectic quasi-states

Chapter 14. Construction by spectral invariant with bulk

14.1. Existence of the limit

14.2. partial quasi-morphism property of ₑ^{\frak }

14.3. Partial symplectic quasi-state property of ^{\frak }ₑ

Chapter 15. Poincaré duality and spectral invariant

15.1. Statement of the result

15.2. Algebraic preliminary

15.3. Duality between Floer homologies

15.4. Duality and Piunikhin isomorphism

15.5. Proof of Theorem 1.1

Chapter 16. Construction of quasi-morphisms via spectral invariant with bulk

Part 4 . Spectral invariants and Lagrangian Floer theory

Chapter 17. Operator \frak

review

Chapter 18. Criterion for heaviness of Lagrangian submanifolds

18.1. Statement of the results.

18.2. Floer homologies of periodic Hamiltonians and of Lagrangian submanifolds

18.3. Filtration and the map \frak _{( , )}^{ ,\frak }

18.4. Identity \frak _{( , )}^{ ,\frak ,∗}∘\CP_{( ᵪ, ),∗}^{\frak }= _{ , }*

18.5. Heaviness of

Chapter 19. Linear independence of quasi-morphisms.

Part 5 . Applications

Chapter 20. Lagrangian Floer theory of toric fibers: review

20.1. Toric manifolds: review

20.2. Review of Floer cohomology of toric fiber

20.3. Relationship with the Floer cohomology in Chapter 17

20.4. Properties of Floer cohomology _{ }(( , )

Λ): review

Chapter 21. Spectral invariants and quasi-morphisms for toric manifolds

21.1. ₑ^{\frak }-heaviness of the Lagrangian fibers in toric manifolds

21.2. Calculation of the leading order term of the potential function in the toric case: review

21.3. Existence of Calabi quasi-morphism on toric manifolds

21.4. Defect estimate of a quasi-morphism ₑ^{\frak }

Chapter 22. Lagrangian tori in -points blow up of \C ² ( ≥2)

Chapter 23. Lagrangian tori in ²× ²

23.1. Review of the construction from [FOOO6]

23.2. Superheaviness of ( )

23.3. Critical values and eigenvalues of ₁( )

Chapter 24. Lagrangian tori in the cubic surface

Chapter 25. Detecting spectral invariant via Hochschild cohomology

25.1. Hochschild cohomology of filtered _{∞} algebra: review

25.2. From quantum cohomology to Hochschild cohomology

25.3. Proof of Theorem 25.1

25.4. A remark

Part 6 . Appendix

Chapter 26. \CP_{( ᵪ, ᵪ),∗}^{\frak } is an isomorphism

Chapter 27. Independence on the de Rham representative of \frak

Chapter 28. Proof of Proposition 20.7

28.1. Pseudo-isotopy of filtered _{∞} algebra

28.2. Difference between \frak ^{ } and \frak

28.3. Smoothing ⁿ-invariant chains

28.4. Wrap-up of the proof of Proposition 3.1.

28.5. Proof of Lemma 3.3

Chapter 29. Seidel homomorphism with bulk

29.1. Seidel homomorphism with bulk

29.2. Proof of Theorem 29.13

29.3. Proof of Theorem 29.9

Chapter 30. Spectral invariants and Seidel homomorphism

30.1. Valuations and spectral invariants

30.2. The toric case

Part 7 . Kuranishi structure and its CF-perturbation: summary

Chapter 31. Kuranishi structure and good coordinate system

31.1. Orbifold

31.2. Kuranishi structure

Chapter 32. Strongly smooth map and fiber product

32.1. Strongly smooth map

32.2. Fiber product

Chapter 33. CF perturbation and integration along the fiber

33.1. Differential form on the space with Kuranishi structure

33.2. CF-perturbation

33.3. Integration along the fiber

Chapter 34. Stokes' theorem

34.1. Normalized boundary

34.2. Statement of Stokes' theorem

Chapter 35. Composition formula

35.1. Smooth correspondence and its perturbation

35.2. Composition of smooth correspondences

35.3. Statement of Composition formula

Bibliography

Index

Back Cover.

Notes:

"July 2019, volume 260, number 1254 (third of 5 numbers)."

Description based on online resource; title from PDF title page (ebrary, viewed October 15, 2019).

Includes bibliographical references and index.

ISBN:

1-4704-5325-8