Recognition of objects through symplectic capacities

TL;DR

This paper proves that generalized symplectic capacities uniquely identify certain symplectic objects, providing a complete invariant in specific categories, and extends recognition results beyond traditional symplectic cases.

Contribution

It establishes that generalized symplectic capacities serve as complete invariants for a broad class of symplectic objects, answering a longstanding open question.

Findings

Generalized symplectic capacities recognize objects in specified symplectic categories.

The result applies to differential form categories, indicating recognition is not exclusive to symplectic geometry.

First recognition result beyond 2D, ellipsoids, and polydiscs in $\

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.