Noncontractible periodic orbits in cotangent bundles and Floer homology
Given a nontrivial free homotopy class of loops in a closed connected Riemannian manifold and a compactly supported time-dependent Hamiltonian on the open unit disk bundle of the cotangent bundle which is bounded above over the zero section by minus the shortest length of periodic geodesics in the given free homotopy class, we prove existence of a 1-periodic orbit representing the given class. The proof shows that the Biran--Polterovich--Salamon (BPS) capacity is finite for every closed connected Riemannian manifold and every free homotopy class of loops. This immediately implies a dense existence theorem for periodic orbits on level hypersurfaces and, consequently, a refined version of the Weinstein conjecture: Existence of a closed characteristic in every nontrivial free homotopy class.