About torsion in the cokernels of the Johnson homomorphisms

The so-called Johnson homomorphisms $({\tau }_k{)}_{k\ge 1}$ embed the graded space associated to the Johnson filtration of a surface with one boundary component into the Lie ring of positive symplectic derivations $D(H)$. In this paper, we show the existence of torsion in the cokernels of the Johnson homomorphisms, for all even degrees, provided the genus is big enough. The Satoh trace $\mathrm{Tr}$ is one of the known obstructions to be in the image of $\tau$. For each degree, we define a map on $\mathrm{Ker}(\mathrm{Tr})\cap D(H)$ whose image is 2-torsion, and which vanishes on the image of $\tau$. We also give a formula to compute these maps.