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PreprintCC BY-NC-SA 4.0arXiv

Relative trisection maps on 4-manifolds

  • Osamu Saeki
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Abstract

We show that for an arbitrary compact orientable smooth $4$--dimensional manifold M with a given open book structure on the boundary ∂ M ≠ ∅, which may be disconnected, there exists a generic smooth map of M onto the $2$--disk which gives a relative trisection on M consistent with the given open book structure on ∂ M. This gives a new proof to the existence of a relative trisection on M. In fact, it turns out that every relative trisection can be obtained in this way. As a byproduct, we also show that M always admits a decomposition into a PALF (positive allowable Lefschetz fibration) over the $2$--disk D2 and a fiber bundle over the circle with a sutured compression body fiber, consistent with the given open book structure on ∂ M and with the surface bundle structure of the PALF over ∂ D2.

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Cite

BibTeXRIS
BibTeX
@misc{saeki2026relative,
  title = {{Relative trisection maps on 4-manifolds}},
  author = {Osamu Saeki},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02494},
  eprint = {2611.02494},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02494},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Relative trisection maps on 4-manifolds
AU  - Osamu Saeki
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02494
UR  - https://arxiv.org/abs/2611.02494
AB  - We show that for an arbitrary compact orientable smooth $4$--dimensional manifold M with a given open book structure on the boundary ∂ M ≠ ∅, which may be disconnected, there exists a generic smooth map of M onto the $2$--disk which gives a relative trisection on M consistent with the given open book structure on ∂ M. This gives a new proof to the existence of a relative trisection on M. In fact, it turns out that every relative trisection can be obtained in this way. As a byproduct, we also show that M always admits a decomposition into a PALF (positive allowable Lefschetz fibration) over the $2$--disk D^2 and a fiber bundle over the circle with a sutured compression body fiber, consistent with the given open book structure on ∂ M and with the surface bundle structure of the PALF over ∂ D^2.
ER  -