A generating set for the Johnson kernel

Let $\Sigma_g \to E \to \Sigma_h$ be a surface bundle over a surface with monodromy representation $\rho: \pi_1 \Sigma_h \to \operatorname{Mod}(\Sigma_g)$ contained in the Torelli group $\mathcal{I}_g$. In this paper we express the cup product structure in $H^*(E, \mathbb{Z})$ in terms of the Johnson homomorphism $\tau: \mathcal{I}_g \to \wedge^3 (H_1 (\Sigma_g, \mathbb{Z}))$. This is applied to the question of obtaining an upper bound on the maximal $n$ such that $p_1: E \to \Sigma_{h_1}, ..., p_n: E \to \Sigma_{h_n}$ are fibering maps realizing $E$ as the total space of a surface bundle over a surface in $n$ distinct ways. We prove that any nontrivial surface bundle over a surface with monodromy contained in the Johnson kernel $\mathcal{K}_g$ fibers in a unique way.

The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. In terms of the representation theory of the symplectic group, we give a complete description of cup products of two classes in the first rational cohomology of the Johnson kernel obtained by the rational dual of the second Johnson homomorphism.

We prove that the first complex homology of the Johnson subgroup of the Torelli group T_g is a non-trivial, unipotent T_g -module for all g\ge 4 and give an explicit presentation of it as a \operatorname{Sym}_\bullet H_1(T_g,\mathbb C) -module when g\ge 6 . We do this by proving that, for a finitely generated group G satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of G . In this setup, we also obtain a precise nilpotence test.

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 $${\text {Mod}}_g$$ and the Torelli group $$\mathcal {I}_g$$ . We show that 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 $$H^{k-2}({\text {Mod}}_{g,*}, \wedge ^k H_1)$$ . 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 $${\text {im}}(\tau _{4i}) \le \wedge ^{4i+2}H_1$$ , 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 $$\mathcal K_{g,*}$$ .

We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various niteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have nite rst Betti number, when the genus is at least 4. We also prove that, in this range, the I-adic completion of the Alexander invariant is nite-dimensional, and the Kahler property for the Torelli group implies the nite generation of the Johnson kernel.

We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping class group.