  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; there
exist stable Hamiltonian structures that are not homotopic to a positive
contact structure; stable Hamiltonian structures are generically Morse-Bott
(i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard
contact structure on the 3-sphere is homotopic to a stable Hamiltonian
structure which cannot be embedded in 4-space. Moreover, we derive a structure
theorem in dimension three and classify stable Hamiltonian structures supported
by an open book. We also discuss implications for the foundations of symplectic
field theory.
