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

Salter, Nick

doi:10.2140/agt.2015.15.3613

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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

DOI: 10.2140/agt.2015.15.3613

Abstract

Let $Σ_{g} \to E \to Σ_{h}$ be a surface bundle over a surface with monodromy representation $ρ : π_{1} Σ_{h} \to Mod (Σ_{g})$ contained in the Torelli group $ℐ_{g}$ . We express the cup product structure in $H^{*} (E, ℤ)$ in terms of the Johnson homomorphism $τ : ℐ_{g} \to \land^{3} (H_{1} (Σ_{g}, ℤ)) ∕ H_{1} (Σ_{g}, ℤ)$ . This is applied to the question of obtaining an upper bound on the maximal $n$ such that $p_{1} : E \to Σ_{h_{1}}, \dots, p_{n} : E \to Σ_{h_{n}}$ 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 $K_{g}$ fibers in a unique way.

Keywords

surface bundles over surfaces, Johnson homomorphism, cup products

Mathematical Subject Classification 2010

Primary: 57R22

Secondary: 57R95

References

Publication

Received: 7 December 2014

Revised: 29 March 2015

Accepted: 6 April 2015

Published: 12 January 2016

Authors

Nick Salter
Department of Mathematics University of Chicago 5734 S University Ave Chicago, IL 60637 USA
http://math.uchicago.edu/~nks/

Volume 15, issue 6 (2015)