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

TL;DR

This paper links cup products in surface bundles to the cohomology of the mapping class group and Torelli group, revealing how certain classes and invariants determine bundle properties and cohomology ring structures.

Contribution

It establishes new connections between MMM classes, Johnson invariants, and surface bundle cohomology, extending invariants and answering longstanding questions.

Findings

Twisted MMM class m_{0,k} computes k-fold cup products.

Higher Johnson invariant τ_{k-2} extends via m_{0,k}.

Surface bundles with monodromy in the Johnson kernel have trivial cohomology rings.

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 $\tau_{k-2}$ 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 $\operatorname{im}(\tau_{4i}) \le \wedge^{4i+2}H_1$, 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 $\tau_i$ when restricted to $\mathcal K_{g,*}$.