Cup products in surface bundles, higher Johnson invariants, and MMM classes (Q1745335)

Let \(\text{Mod}_g\) be the mapping class group of a closed oriented surface \(\Sigma_g\) of genus \(g \geq 2\). The Mumford-Morita-Miller classes are cohomology classes \[ e_{i} \in H^{2i}(\text{Mod}_g), \qquad i \geq 1 \] which, in virtue of the Madsen-Weiss theorem, generate the stable cohomology of \(\text{Mod}_g\). (Here and below, we assume coefficients in \(\mathbb{Q}\).) \textit{N. Kawazumi} introduced in [Invent. Math. 131, No. 1, 137--149 (1998; Zbl 0894.57020)] a ``twisted'' version of the MMM classes: \[ m_{i,j} \in H^{2i+j-2}(\text{Mod}_{g,*};\Lambda^j H), \qquad i,j \geq 0 \] where \(\text{Mod}_{g,*}\) denotes the mapping class group of the surface \(\Sigma_g\) with a marked point \(*\) and \(H:= H_1(\Sigma_{g})\). Specifically, \(m_{i,0}\) coincides with \(e_{i-1}\) for \(i\geq 2\). The paper under review contains several results about the other ``extreme'' twisted MMM classes, namely the classes \(m_{0,j}\) with \(j\geq 2\). The authors define in \(H^{j-2}(B;\Lambda^j H)\) characteristic classes of \(\Sigma_g\)-bundles \(E\to B\) with section (here the fundamental group of the base space \(B\) acts on \(H\) via the group homomorphism \(\pi_1(B) \to \text{Mod}_{g,*}\) that classifies the surface bundle): then, the first result is a computation of some iterated cup products in the total space \(E\) using those characteristic classes. Next, the author considers the restriction of the same classes \[ m_{0,j} \in H^{j-2}( \mathcal{I}_{g,*};\Lambda^j H) \] to the Torelli group \(\mathcal{I}_{g,*}\) which, by definition, is the subgroup of \(\text{Mod}_{g,*}\) acting trivially on \(H\); as a second result, it is proved that \(m_{0,j}\) coincides with (a multiple of) the linear map \[ \tau_{j-2}: H_{j-2}(\mathcal{I}_{g,*}) \longrightarrow \Lambda^j H \] that \textit{D. Johnson} defined in his survey paper [Contemp. Math. 20, 165--179 (1983; Zbl 0553.57002)] using Jacobian varieties. After that, the author switches from the closed surface \(\Sigma_g\) with marked point to the surface \(\Sigma_{g,1}\) with one boundary component. In this setting, he links the non-triviality of the MMM class \(e_{k}\) on the Torelli group \(\mathcal{I}_{g,1}\) to the existence of a trivial \(\mathfrak{sp}_{2g}\)-representation inside the image of \(\tau_{2k}\), and he deduces that \(\tau_{4k-2}\) is not surjective; this partly answers a question raised by Johnson in the same survey paper. The author concludes with an interesting application: any \(\Sigma_g\)-bundle \(E\to B\) such that \(\pi_1(B)\) acts trivially on the second nilpotent quotient \(\pi/[\pi,[\pi,\pi]]\) of \(\pi:=\pi_1(\Sigma_g)\) has the same cohomology algebra as the trivial bundle \(B \times \Sigma_g\). (When \(B=S^1\), this directly follows from the equivalence between two possible definitions of the ``classical'' Johnson homomorphism, which is \(\tau_1\) in the above notation.)