Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 

 

Sections of surface bundles and Lefschetz fibrations


Authors: R. İnanç Baykur, Mustafa Korkmaz and Naoyuki Monden
Journal: Trans. Amer. Math. Soc. 365 (2013), 5999-6016
MSC (2010): Primary 57R22, 57R17, 20F65
Published electronically: August 2, 2013
MathSciNet review: 3091273
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Abstract: We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus $ g$ and the base genus $ h$ are positive, we prove that the adjunction bound $ 2h-2$ is the only universal bound on the self-intersection number of a section of any such genus $ g$ bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is $ 1/2$. We furthermore prove that there is no upper bound on the number of critical points of genus-$ g$ Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for $ g \geq 2$.


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R. İnanç Baykur
Affiliation: Max Planck Institut für Mathematik, Bonn, Germany – and – Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453
Email: baykur@mpim-bonn.mpg.de, baykur@brandeis.edu

Mustafa Korkmaz
Affiliation: Department of Mathematics, Middle East Technical University, Ankara, Turkey
Email: korkmaz@metu.edu.tr

Naoyuki Monden
Affiliation: Department of Mathematics, Osaka University, Osaka, Japan
Email: n-monden@cr.math.sci.osaka-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05840-0
Received by editor(s): October 26, 2011
Received by editor(s) in revised form: February 10, 2012, March 6, 2012, and March 11, 2012
Published electronically: August 2, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.