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 $(M,\omega)$, such that $M$ is a compact and 1-connected manifold, $\omega$ 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 $\mathbb{R}^4$. 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.