Octav Cornea And

Contents Introduction. 1 1. Overview of the main constructions and results. 4 1.1. Definition and properties of the cluster complex 4 1.2. Fine Floer Homology. 9 1.3. Symmetrization. 15 1.4. Applications. 18 2. The cluster complex and its homology. 20 2.1. Clustered moduli spaces 20 2.2. Invariance of the cluster homology. 30 2.3. Various special cases. 35 3. The fine Floer complex and its homology 37 3.1. Moduli spaces for the fine Floer Homology. 37 3.2. Invariance of the fine Floer Homology. 40 4. Some applications and comments. 42 4.1. The symmetric fine Floer complex and the cluster complex 42 4.2. The cluster complex in minimal form 44 4.3. Detection of disks of bounded area 45 4.4. Some calculations. 47 4.5. Detection of closed orbits of Hamiltonian systems 48 References 55 Introduction. We introduce in this paper a new homology attached to a Langrangian subman-ifold L of a symplectic manifold, which provides an invariant of the Hamiltonian isotopy class of L. This leads to various applications concerning the analytical, topological, and dynamical properties of Lagrangian submanifolds, and to a new universal Floer homology, defined without obstruction, for pairs of Lagrangian sub-manifolds. Throughout this paper, we shall assume that all Lagrangian submanifolds are connected, orientable, and relatively spin (recall that a Lagrangian submanifold