Sections of surface bundles and Lefschetz fibrations

Baykur, R.; Korkmaz, Mustafa; Monden, Naoyuki

doi:10.1090/S0002-9947-2013-05840-0

Sections of surface bundles and Lefschetz fibrations
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by R. İnanç Baykur, Mustafa Korkmaz and Naoyuki Monden PDF

Trans. Amer. Math. Soc. 365 (2013), 5999-6016 Request permission

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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