Cup products, the Johnson homomorphism and surface bundles over surfaces with multiple fiberings

Abstract

Let ${\Sigma }_g\to E\to {\Sigma }_h$ be a surface bundle over a surface with monodromy representation $\rho :{\pi }_1{\Sigma }_h\to Mod\left({\Sigma }_g\right)$ contained in the Torelli group ${\mathcal{ℐ}}_g$. We express the cup product structure in $H^{\ast }\left(E,\mathbb{Z}\right)$ in terms of the Johnson homomorphism $\tau :{\mathcal{ℐ}}_g\to {\wedge }^3\left(H_1\left({\Sigma }_g,\mathbb{Z}\right)\right)∕H_1\left({\Sigma }_g,\mathbb{Z}\right)$. This is applied to the question of obtaining an upper bound on the maximal $n$ such that $p_1:E\to {\Sigma }_{h_1},\dots ,p_n:E\to {\Sigma }_{h_n}$ 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.