Image of the third Johnson homomorphism

In this note we show that the image of the third Johnson homomorphism ${\tau }_3$ coincides with the kernel of Morita’s trace map for the case of a surface of genus $g$ with one boundary component ${\mathrm{\Sigma }}_{g,1}$, when $g\ge 6$. Moreover, as a consequence, we get that the image of ${\tau }_3$ on the handlebody subgroups coincides with the intersection of kernels of the Morita trace map and the Lagrangian trace maps.