Floer homology and the symplectic isotopy problem

The symplectic isotopy problem is a question about automorphisms of a compact symplectic manifold. It asks whether the relation of sym-plectic isotopy between such automorphism is finer than the relation of diffeotopy (smooth isotopy). The principal result of this thesis is that there are symplectic manifolds for which the answer is positive; in fact, a large class of symplectic four-manifolds is shown to have this property. This result is the consequence of the study of a special class of symplectic automorphisms, called generalized Dehn twists. The hard part of studying the symplectic isotopy problem is how to prove that two given symplectic automorphisms are not symplectically isotopic. Symplectic Floer homology theory assigns a ‘homology group’ to any symplectic automorphism. These groups are invariant under symplectic isotopy, hence an obvious candidate for the task. While there is no general procedure for computing the Floer homology groups