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Sections, pseudosections, and multisections of Lefschetz fibrations
Abstract
We show that not every symplectic Lefschetz fibration over the $2$--sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over S2 admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus g≥ 7, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.
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BibTeX
@misc{baykur2026sections,
title = {{Sections, pseudosections, and multisections of Lefschetz fibrations}},
author = {R. Inanc Baykur and Noriyuki Hamada},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01776},
eprint = {2610.01776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01776},
}
RIS
TY - UNPB
TI - Sections, pseudosections, and multisections of Lefschetz fibrations
AU - R. Inanc Baykur
AU - Noriyuki Hamada
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.01776
UR - https://arxiv.org/abs/2610.01776
AB - We show that not every symplectic Lefschetz fibration over the $2$--sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over S^2 admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus g≥ 7, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.
ER -