Cup Products, the Johnson Homomorphism, and Surface Bundles over Surfaces with Multiple Fiberings
نویسنده
چکیده
Let Σg → E → Σh be a surface bundle over a surface with monodromy representation ρ : π1Σh → Mod(Σg) contained in the Torelli group Ig. In this paper we express the cup product structure in H∗(E,Z) in terms of the Johnson homomorphism τ : Ig → ∧(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.
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