The Arnold conjecture, Floer homology and symplectic homology

Abstract

In the sixties V.I. Arnold announced several seminal and fruitful conjectures in symplectic topology. This chapter is devoted to his conjecture about fixed points of symplectic mappings which originates in questions of celestial mechanics dating back to the turn of the century. Reformulated dynamically, this conjecture gives a lower bound for the number of global forced oscillations of a time-dependent Hamiltonian vector field on a compact symplectic manifold in terms of the topology of this manifold. Forced oscillations are singled out by the action principle; solutions of the conjecture gave rise to new ideas and techniques in the calculus of variations and in nonlinear elliptic systems. In particular, it prompted A. Floer 1986 to the construction of his homology theory.