Geodesic
Cup products, the Johnson homomorphism and surface bundles over surfaces with multiple fiberings
Salter, Nick1
1Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA
Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3613-3652
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Let Σg → E → Σh be a surface bundle over a surface with monodromy representation ρ: π1Σh → Mod(Σg) contained in the Torelli group ℐg. We express the cup product structure in H∗(E, ℤ) in terms of the Johnson homomorphism τ: ℐg →∧ 3(H1(Σg, ℤ))∕H1(Σg, ℤ). This is applied to the question of obtaining an upper bound on the maximal n such that p1: E → Σh1,…,pn: E → Σhn 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 Kg fibers in a unique way.

DOI : 10.2140/agt.2015.15.3613
Classification : 57R22, 57R95
Keywords: surface bundles over surfaces, Johnson homomorphism, cup products

Salter, Nick  1

1 Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA 
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Salter, Nick. Cup products, the Johnson homomorphism and surface bundles over surfaces with multiple fiberings. Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3613-3652. doi: 10.2140/agt.2015.15.3613

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