Trisections of non-orientable 4-manifolds

Miller, Maggie; Naylor, Patrick

doi:10.48550/arxiv.2010.07433

Cited by 1 publication

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“…Remark 2.11. In [27], the third author and Naylor showed that a smooth, closed, non-orientable 4-manifold X 4 is also determined up to diffeomorphism by (framed) attaching regions of 0, 1, and 2-handles. If desired, one could thus make sense of diagrams of closed (immersed) surfaces in Kirby diagrams of non-orientable 4manifolds.…”

Section: 23mentioning

confidence: 99%

Band diagrams of immersed surfaces in 4-manifolds

Hughes¹,

Kim²,

Miller³

2021

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We study immersed surfaces in smooth 4-manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4-manifold, representing a level set of the surface with respect to an associated Morse function. We show that every self-transverse immersed surface in a smooth, orientable, closed 4-manifold can be represented by a singular banded unlink diagram, and that such representations are uniquely determined by the ambient isotopy or equivalence class of the surface up to a set of singular band moves which we define explicitly. By introducing additional finger, Whitney, and cusp diagrammatic moves, we can use these singular band moves to describe homotopies or regular homotopies as well.Using these techniques, we introduce bridge trisections of immersed surfaces in arbitrary trisected 4-manifolds and prove that such bridge trisections exist and are unique up to simple perturbation moves. We additionally give some examples of how singular banded unlink diagrams may be used to perform computations or produce explicit homotopies of surfaces.

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Section: 23mentioning

confidence: 99%