The Poisson bracket invariant for open covers consisting of topological disks on surfaces (Q6150158)

Let \((M,\omega)\) be a connected symplectic manifold. Let us recall that a subset \(X\) is displaceable if there exists a Hamiltonian diffeomorphim \(\phi\) such that \(\phi(X)\) is disjoint with \(X\). Let us also recall that the strong bracket conjecture of \textit{L. Polterovich} [Lett. Math. Phys. 102, No. 3, 245--264 (2012; Zbl 1261.81078); Commun. Math. Phys. 327, No. 2, 481--519 (2014; Zbl 1291.81198)] has been proved in dimension 2 by \textit{L. Buhovsky} et al. [Comment. Math. Helv. 95, No. 2, 247--278 (2020; Zbl 1454.53063)]. In this paper, the authors replace the open covers consisting of displaceable sets by open covers consisting of topological disks. The authors give a necessary and sufficient condition for the Poisson bracket to be positive.