First steps in stable Hamiltonian topology

In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on S^3 is homotopic to a stable Hamiltonian structure which cannot be embedded in R^4 . Moreover, we derive a structure theorem for stable Hamiltonian structures in dimension three, study sympectic cobordisms between stable Hamiltonian structures, and discuss implications for the foundations of symplectic field theory.