  We study global transverse Poincar\'{e} sections and give topological
conditions for their existence, showing they never exist in many important
cases. We prove that an energy hypersurface possessing global transverse
Poincar\'{e} section is equivalent to the hypersuface having a cosymplectic
structure. We give a family of Hamiltonian systems with global Poincar\'{e}
sections of all possible topologies. Finally, we address the question of when a
compact hypersurface of a symplectic manifold possesses an induced cosymplectic
structure.
