The full power of Lagrangian Floer homology theory can be mustered only using the A∞ machinery introduced by Fukaya (Fu93) and fully developed in the book (FOOO09). This theory is necessary to deal with general Lagrangian submanifolds when the structure of disc-bubbles is not as simple as in the following two special cases:

1. (1) exact Lagrangian submanifolds in (non-compact) exact symplectic Manifolds

2. (2) monotone Lagrangian submanifolds in (monotone) symplectic manifolds.

Since the A∞ machinery goes beyond the scope of this book, we will focus on these two cases and use them to illustrate the usage of Floer homology in the study of symplectic topology.

However, even when the Floer homology HF(L0, L1; M) is defined it is a highly non-trivial task to explicitly compute this homology as soon as we go beyond the exact case and L1 = φ(L0) for a Hamiltonian diffeomorphism φ.

Theorem 16.4.10 or its cousins is the basic starting point of the application of Floer homology to the study of symplectic topology. Most applications so far are related to the construction of nondisplaceable Lagrangian submanifolds or the study of the symplectic topology of displaceable Lagrangian submanifolds such as the Hofer displacment energy and the Maslov class obstruction. In this study, it is also crucial to analyze the structure of the Floer moduli spaces when φ → id, more precisely under the adiabatic limit of φ(L) → L, which gives rise to thick–thin decomposition of Floer trajectories.

We refer those who are interested in learning more about Lagrangian Floer theory beyond the above two cases and its application to mirror symmetry and symplectic topology to (CO06), (FOOO10b)–(FOOO13).

Nearby Lagrangian pairs: thick–thin dichotomy

In this section, we study some convergence results of M(L, φ(L); p, q) as φ → id in C1. We will also explain how this study of degeneration gives rise to a spectral sequence introduced in (Oh96b). This spectral sequence in this form has been further explored in (Bu10), (BCo09), (D09). A more general version of spectral sequences of this kind is presented in (FOOO09), which also handles non-monotone Lagrangian submanifolds with an additional unobstructedness hypothesis.

Here is the precise setting of the study of this degeneration. Let L be a given compact Lagrangian submanifold. The following notion was introduced in (Oh05d, Spa08).