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
natural geometric conditions in which relative symplectic cohomology of two
subsets satisfies the Mayer–Vietoris property. These conditions involve certain
integrability assumptions involving geometric objects called barriers — roughly, a
–parameter
family of rank
coisotropic submanifolds. The proof uses a deformation argument in which the
topological energy zero (ie constant) Floer solutions are the main actors.
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