Sections of surface bundles and Lefschetz fibrations
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 at least two.
Keywords
Cite
@article{arxiv.1110.1224,
title = {Sections of surface bundles and Lefschetz fibrations},
author = {R. Inanc Baykur and Mustafa Korkmaz and Naoyuki Monden},
journal= {arXiv preprint arXiv:1110.1224},
year = {2012}
}
Comments
19 pages, 2 figures, minor revisions for publication