Recognition of objects through symplectic capacities

Abstract

We prove that the generalized symplectic capacities recognize objects in symplectic categories whose objects are of the form (M,ω), such that M is a compact and 1-connected manifold, ω is an exact symplectic form on M, and there exists a boundary component of M with negative helicity. The set of generalized symplectic capacities is thus a complete invariant for such categories. This answers a question by Cieliebak, Hofer, Latschev, and Schlenk. It appears to be the first result concerning this question, except for recognition results for manifolds of dimension 2, ellipsoids, and polydiscs in R4. Strikingly, our result holds more generally for differential form categories. Recognition of objects is therefore not a symplectic phenomenon.We also prove a version of the result for normalized capacities.