On the group-homological description of the second Johnson homomorphism (Q5957954)

Let \(\Sigma_{g,1}\) be a compact orientable surface of genus \(g\) with one boundary component and \({\mathcal{K}}_{g,1}\) denote the subgroup of the mapping class group of \(\Sigma_{g,1}\) generated by Dehn twists about separating simple closed curves. In [Contemp. Math. 20, 165--179 (1983; Zbl 0553.57002)], \textit{D. L. Johnson} constructed a series of homomorphisms \(\tau_k\) from certain subgroups of the mapping class groups into some abelian groups. Then in [Duke Math. J. 70, 699--726 (1993; Zbl 0801.57011)], \textit{S. Morita} constructed refinements \(\tilde{\tau}_k\) of \(\tau_k\) such that \(d^2\circ\tilde{\tau}_k = \tau_k\) for some natural homomorphism \(d^2\). The homomorphisms \(\tau_2\) and \(\tilde{\tau}_2\) are defined on \({\mathcal{K}}_{g,1}\) giving abelian quotients. The author proves in this paper that the map \(d^2\) restricts to an isomorphism between the images of \(\tilde{\tau}_2\) and \(\tau_2\), concluding that \(\tilde{\tau}_2\) gives no new abelian quotient of \({\mathcal{K}}_{g,1}\).