Locality of relative symplectic cohomology for complete embeddings

Abstract

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.

On certain quantifications of Gromov’s nonsqueezing theorem

“…We now establish a correspondence between the rational homology sphere B 0 a and the space S a;q coming from the smoothing in Lemma A. 10. Define yet another copy † 00 a of the lens space by setting † 00 a D 1 a .…”