  In this paper, we construct a Hamiltonian Floer theory based invariant called
relative symplectic cohomology, which assigns a module over the Novikov ring to
compact subsets of closed symplectic manifolds. We show the existence of
restriction maps, and prove some basic properties. Our main contribution is to
identify a natural geometric situation in which relative symplectic cohomology
of two subsets satisfy the Mayer-Vietoris property. This is tailored to work
under certain integrability assumptions, the weakest of which introduces a new
geometric object called a barrier - roughly, a one parameter family of rank 2
coisotropic submanifolds. The proof uses a deformation argument in which the
topological energy zero (i.e. constant) Floer solutions are the main actors.
