### 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.

Original language | English |
---|---|

Pages (from-to) | 5999-6016 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 365 |

Issue number | 11 |

DOIs | |

Publication status | Published - Aug 27 2013 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*365*(11), 5999-6016. https://doi.org/10.1090/S0002-9947-2013-05840-0

**Sections of surface bundles and lefschetz fibrations.** / Baykur, R. Inanç; Korkmaz, Mustafa; Monden, Naoyuki.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 365, no. 11, pp. 5999-6016. https://doi.org/10.1090/S0002-9947-2013-05840-0

}

TY - JOUR

T1 - Sections of surface bundles and lefschetz fibrations

AU - Baykur, R. Inanç

AU - Korkmaz, Mustafa

AU - Monden, Naoyuki

PY - 2013/8/27

Y1 - 2013/8/27

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84882621099&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84882621099&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2013-05840-0

DO - 10.1090/S0002-9947-2013-05840-0

M3 - Article

AN - SCOPUS:84882621099

VL - 365

SP - 5999

EP - 6016

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -