Symplectic topology and ideal-valued measures

Abstract

We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgunes.

Heavy sets and index bounded relative symplectic cohomology

“…Theorem 1.2 (Corollary 1.55 [4]). If (M, ω) is symplectically aspherical and K is a heavy contact-type region with incompressible index bounded boundary, then K is SH-heavy.…”

“…Remark 1.4. We remark that our definition of the index bounded condition is slightly different than those in [4] and [20]. See (2.6).…”

“…This idea of distributing the quantum cohomology ring to compact sets was further explored by Dickstein-Ganor-Polterovich-Zapolsky [4]. They successfully combined the theory of the relative symplectic cohomology with the theory of idealvalued measures, to construct a symplectic ideal-valued quasi-measure on any closed symplectic manifold.…”

“…Conjecture 1.1 (Conjecture 1.52 [4]). A compact subset of a closed symplectic manifold is heavy if and only if it is SH-heavy.…”

Index bounded relative symplectic cohomology

“…One motivation of it comes from mirror symmetry suggested by Seidel [23] and the family Floer program. On the other hand, some symplectic topological applications have already appeared in [25,9]. Also see its relation with quantum cohomology by Borman-Sheridan-Varolgunes [3].…”