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A complete embedding is a symplectic embedding 
$\iota :Y\to M$ of a geometrically bounded symplectic manifold 
$Y$ into another geometrically bounded symplectic manifold 
$M$ of the same dimension. When 
$Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset 
$K$ inside 
$Y$ is naturally isomorphic to that of its image 
$\iota (K)$ inside 
$M$. Under the assumption that the torsion exponents of 
$K$ are bounded, we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions.
