Spectral invariants with bulk, quasi-morphisms and Lagrangian Floer theory

http://www.ams.org/books/memo/1254/

스펙트럼 불변량(spectral invariants) 이론은 해밀턴 동역학(Hamilton dynamics)과 사교위상 영역에서  플로어 호몰로지를 이용하여 만들어진 정량적 불변량 중 단연코 가장 강력하며 유용한 이론이디. 266쪽에 달하는 이 논문은 라그랑지언 플로어 이론, 열린 그로모프-위튼과 스펙트럼 불변량이론을 이를 저자들이 이전에 발전시킨 옹골찬 토릭 다양체의 라그랑지안 플로어 이론과 란다우-긴즈버그 모델과의 거울대칭의 쌍대성 증명에 사용되었던 고차 범주 이론의 호몰로지 대수이론의 언어로 결합하고 새로운 사교 준 상태(symplectic quasi-state), 준 모피즘(quasi-morphism)의 구성과 그리고 라그랑지안 교차점에 대한 새로운 결과등 사교 위상 수학의 많은 문제에 적용하여 사교위상의 풀리지 않았던 여러 문제들에 대한 답하였다. 이 논문에서 사용된 방법론은 향 후 사교 위상수학 발전에 시금석이 될 것으로 기대된다.

 

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 manifoldsThe most novel part of this paper is to use open-closed Gromov-Witten-Floer theory (operator 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 which admits uncountably many independent quasi-morphisms . We also obtain a new intersection result for the Lagrangian submanifold in discovered in Fukaya, et al. (2012).Many of these applications were announced in Fukaya, et al. (2010, 2011, 2012).