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 $\rho :{\pi }_1{\Sigma }_h\to \mathrm{Mod}({\Sigma }_g)$ contained in the Torelli group ${\mathcal{I}}_g$. In this paper we express the cup product structure in $H^{\ast }(E,\mathbb{Z})$ 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.

This version contains an updated introduction, incorporating discussion of our recent work [arXiv:1407.2062]. The organization has been revised and material from the old Section 2 has been split into three separate sections. There are various other miscellaneous improvements to the exposition. 33 pages with 2 figures

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