The Johnson homomorphism and the third rational cohomology group of the Torelli group (Q1767721)

The Torelli group \(\mathcal{I}_g\) is defined by taking the mapping class group of a closed oriented surface of genus \(g\) at least \(3\); \(\mathcal{I}_g\) consists of those elements of the mapping class group which act trivially on the first homology group, \(H\), of the surface. \textit{D. Johnson} [Math. Ann. 249, 225--242 (1980; Zbl 0409.57009)] defined a homomorphism with domain \(\mathcal{I}_g\) and range a three-fold product of \(H\) modulo a certain subgroup isomorphic to \(H\). Following \textit{D. Johnson} [Topology 24, 127--144 (1985; Zbl 0571.57010)] and \textit{R. Hain} [J. Am. Math. Soc. 10, 597--651 (1997; Zbl 0915.57001)], the author uses the induced homomorphism on homology to study the third rational homology group of \(\mathcal{I}_g\).