Volume 15, issue 6 (2015)

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

Nick Salter

Algebraic & Geometric Topology 15 (2015) 3613–3652
Abstract

Let ${\Sigma }_{g}\to E\to {\Sigma }_{h}$ be a surface bundle over a surface with monodromy representation $\rho :\phantom{\rule{0.3em}{0ex}}{\pi }_{1}{\Sigma }_{h}\to Mod\left({\Sigma }_{g}\right)$ contained in the Torelli group ${\mathsc{ℐ}}_{g}$. We express the cup product structure in ${H}^{\ast }\left(E,ℤ\right)$ in terms of the Johnson homomorphism $\tau :\phantom{\rule{0.3em}{0ex}}{\mathsc{ℐ}}_{g}\to {\wedge }^{3}\left({H}_{1}\left({\Sigma }_{g},ℤ\right)\right)∕{H}_{1}\left({\Sigma }_{g},ℤ\right)$. This is applied to the question of obtaining an upper bound on the maximal $n$ such that ${p}_{1}:\phantom{\rule{0.3em}{0ex}}E\to {\Sigma }_{{h}_{1}},\dots ,{p}_{n}:\phantom{\rule{0.3em}{0ex}}E\to {\Sigma }_{{h}_{n}}$ 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 ${\mathsc{K}}_{g}$ fibers in a unique way.

Keywords
surface bundles over surfaces, Johnson homomorphism, cup products
Primary: 57R22
Secondary: 57R95