Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism

Abstract

Given a closed connected symplectic manifold $(M,ω)$, we construct an alternating $\mathbb{R}$-bilinear form $\mathfrak{b}=\mathfrak{b}_{μ_{\mathrm{Sh}}}$ on the real first cohomology of $M$ from Shelukhin's quasimorphism $μ_{\mathrm{Sh}}$. Here $μ_{\mathrm{Sh}}$ is defined on the universal cover of the group of Hamiltonian diffeomorphisms on $(M,ω)$. This bilinear form is invariant under the symplectic mapping class group action, and $\mathfrak{b}$ yields a constraint on the fluxes of commuting two elements in the group of symplectomorphisms on $(M,ω)$. These results might be seen as an analog of Rousseau's result for an open connected symplectic manifold, where he recovered the symplectic pairing from the Calabi homomorphism. Furthermore, $\mathfrak{b}$ controls the extendability of Shelukhin's quasimorphisms, as well as the triviality of a characteristic class of Reznikov. To construct $\mathfrak{b}$, we build general machinery for a group $G$ of producing a real-valued $\mathbb{Z}$-bilinear form $\mathfrak{b}_μ$ from a $G$-invariant quasimorphism $μ$ on the commutator subgroup of $G$.