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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 Σg E Σh be a surface bundle over a surface with monodromy representation ρ: π1Σh Mod(Σg) contained in the Torelli group g. We express the cup product structure in H(E, ) in terms of the Johnson homomorphism τ: g 3(H1(Σg, ))H1(Σg, ). 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.

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/