### Abstract

We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M × S^{1}where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory.

Original language | English |
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Pages (from-to) | 674-724 |

Number of pages | 51 |

Journal | Proceedings of the London Mathematical Society |

Volume | 113 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2016 Jan 1 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the London Mathematical Society*,

*113*(5), 674-724. https://doi.org/10.1112/plms/pdw042

**Elimination of cusps in dimension 4 and its applications.** / Behrens, Stefan; Hayano, Kenta.

Research output: Contribution to journal › Article

*Proceedings of the London Mathematical Society*, vol. 113, no. 5, pp. 674-724. https://doi.org/10.1112/plms/pdw042

}

TY - JOUR

T1 - Elimination of cusps in dimension 4 and its applications

AU - Behrens, Stefan

AU - Hayano, Kenta

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M × S1where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory.

AB - We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M × S1where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory.

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U2 - 10.1112/plms/pdw042

DO - 10.1112/plms/pdw042

M3 - Article

VL - 113

SP - 674

EP - 724

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 5

ER -