ABSTRACT
This chapter proposes a natural generalization of the celebrated h-cobordism and S-cobordism theorems to loop space. It explains part of the idea of the proof of the half, and also of a finite-dimensional analogue of the conjecture which can be proved under some additional assumptions. The chapter reviews the famous results due to Smale et al. in differential topology. In the case when there is a fundamental group, there is one more obstruction: the Whitehead torsion. J. Eliashberg and M. Gromov developed a symplectic analogue of pseudo-isotopy theory for Lagrangian intersections in the cotangent bundle by using generating functions. The chapter discusses a symplectic analogue of the h-cobordism and S-cobordism theorems by using Floer homology. It also discusses the non-simply-connected case. The chapter explores that any deformation of a Lagrangian submanifold is realized by an exact symplectic diffeomorphism if and only if the Lagrangian submanifold is simply connected.
