SECTIONS OF SURFACE BUNDLES AND LEFSCHETZ FIBRATIONS


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Baykur R. I. , KORKMAZ M., Monden N.

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.365, no.11, pp.5999-6016, 2013 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 365 Issue: 11
  • Publication Date: 2013
  • Doi Number: 10.1090/s0002-9947-2013-05840-0
  • Journal Name: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.5999-6016

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 >= 2.