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

Abstract

In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group $\operatorname{Mod}_g$ and the Torelli group $\mathcal{I}_g$. We show that N. Kawazumi's twisted MMM class $m_{0,k}$ can be used to compute $k$-fold cup products in surface bundles, and that $m_{0,k}$ provides an extension of the higher Johnson invariant $τ_{k-2}$ to $H^{k-2}(\operatorname{Mod}_{g,*}, \wedge^k H_1)$. These results are used to show that the behavior of the restriction of the even MMM classes $e_{2i}$ to $H^{4i}(\mathcal{I}_g^1)$ is completely determined by $\operatorname{im}(τ_{4i}) \le \wedge^{4i+2}H_1$, and to give a partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the Johnson kernel $\mathcal K_{g,*}$ have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all $τ_i$ when restricted to $\mathcal K_{g,*}$.