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The Quarterly Journal of Mathematics 2001 52(2):217-228; doi:10.1093/qjmath/52.2.217
© 2001 by Oxford University Press
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Geometric Monodromy and the Hyperbolic Disc

Ivan Smith1

1 New College, Oxford OX1 3BN, UK

Symplectic four-manifolds give rise to Lefschetz fibrations, which are determined by monodromy representations of free groups in mapping class groups. We study the topology of Lefschetz fibrations by analysing the action of the monodromy on the universal cover of a smooth fibre and give a new and simple proof that Lefschetz fibrations arising from Donaldson's construction via pencils of sections never decompose as non-trivial fibre sums; in particular not all Lefschetz fibrations are fibre sums of holomorphic Lefschetz fibrations. We also show that there can never be isotopy classes of simple closed curve invariant under the monodromy and as a corollary we give a symplectic analogue of Manin's theorem, showing that Lefschetz fibrations admit at most finitely many homotopy classes of geometric section.


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Home page
 Int Math Res NoticesHome page
M. Korkmaz
Lefschetz Fibrations and an Invariant of Finitely Presented Groups
Int Math Res Notices, January 21, 2009; (2009) rnn164v1.
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