# The Poisson bracket invariant for open covers consisting of topological disks on surfaces

## TL;DR

This paper investigates the Poisson bracket invariant for open covers of surfaces by topological disks, establishing a necessary and sufficient condition for the invariant to be positive, extending previous work on displaceable sets.

## Contribution

It provides a new characterization of Poisson bracket invariants for covers of surfaces by topological disks, broadening the scope beyond displaceable sets.

## Key findings

- Established a necessary and sufficient condition for positivity of Poisson bracket invariants.
- Extended previous results from displaceable sets to topological disks.
- Contributed to the understanding of Poisson bracket invariants in symplectic topology.

## Abstract

L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension $2$. In this note, instead of open cover consisting of displaceable sets in their work, we consider open cover constituted of topological discs and give a necessary and sufficient condition that Poisson bracket invariants of these covers are positive.