An affirmative answer to a conjecture on the Metoki class

Abstract

In [6], Kotschick and Morita showed that the Gel'fand–Kalinin–Fuks class in ${\mathrm{H}^{7}_{\rm GF} ( {\mathfrak{ham}}_{2}, {\mathfrak{sp}}(2,\mathbb{R}))_{8}}$ is decomposed as a product $\eta \wedge \omega$ of some leaf cohomology class $\eta$ and a transverse symplectic class $\omega$. We show that the same formula holds for the Metoki class, which is a non-trivial element in ${\mathrm{H}^{9}_{\rm GF} ( {\mathfrak{ham}}_{2}, {\mathfrak{sp}}(2,\mathbb{R}))_{14}}$. The result was conjectured in [6], where they studied characteristic classes of transversely symplectic foliations due to Kontsevich. Our proof depends on Gröbner Basis theory using computer calculations.