Trisections of $4$-manifolds via Lefschetz fibrations

Castro, Nickolas A.; Özbağcı, Burak

doi:10.4310/mrl.2019.v26.n2.a3

Cited by 22 publications

(14 citation statements)

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“…Castro [1] developed this case further, in particular demonstrating the importance of this case by showing how to glue two trisected 4-manifolds along a common boundary to produce a trisection of a closed manifold. The current authors then worked out [2] the diagrammatic version of this theory, with the diagrammatic version of gluing appearing in Castro's work with Ozbagci [3]. This paper supplements these papers by first showing, through examples and a careful exposition of the existence proof in [4], how to explicitly turn a handle decomposition of a 4manifold together with the data of a given open book on the boundary into a trisection of the 4-manifold described diagrammatically.…”

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confidence: 91%

Trisections of 4-manifolds with boundary

Castro

Pinzón-Caicedo²

2018

Proc. Natl. Acad. Sci. U.S.A.

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Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4manifolds with multiple boundary components, and show how trisected 4-manifolds with multiple boundary components glue together.

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confidence: 91%

Trisections of 4-manifolds with boundary

Castro

Pinzón-Caicedo²

2018

Proc. Natl. Acad. Sci. U.S.A.

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“…The three resulting vanishing cycles of the indefinite folds are characterized by the property that they each intersect transversely once positively and that any arc from one boundary component of the surface to the other must intersect one of the curves. We note that the effect of wrinkling on relative trisection diagrams is studied in [5].…”

Section: Reference Paths and Modifications Of Wrinkled Fibrationsmentioning

confidence: 99%

Uniqueness of 4-manifolds described as sequences of 3-d handlebodies

Islambouli¹

2021

Preprint

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Given a description of a manifold, two natural questions arise: existence and uniqueness. For example, closed, orientable 3-manifolds can be described as surgeries on a link in S 3 . The existence problem for this description was addressed by Lickorish and Wallace [21] [29] who showed that any closed, orientable 3-manifold is given by surgery on a link in S 3 . In [15], Kirby proved a uniqueness theorem for these descriptions, showing that any two link-surgeries yielding the same 3-manifold are related by handle slides and blowups. Similarly, Moise [24] showed that every closed, orientable 3-manifold can be described as a Heegaard splitting, and Reidemeister and Singer [26] [25] showed that any two Heegaard splittings giving the same 3-manifold are related by a stabilization operation.Moving up a dimension, there has been a variety of new descriptions of smooth, orientable, closed 4-manifolds. In [8], Gay showed that any such 4manifold can be described as a loop of Morse functions on a surface, and used this fact to give a novel proof of that the smooth, oriented, 4-dimensional cobordism group is isomorphic to Z. In [17], Klug and the author showed that these 4-manifolds can be represented as a loop in the pants complex, and this was used to provide a combinatorial calculation of the aforementioned cobordism group as well as to give invariants of loops in the pants complex. Kirby and Thompson [16] have also showed that any such 4-manifold can be described as a loop in the cut complex, and they used this to define an invariant of 4-manifolds which detects the 4-sphere among homotopy spheres. Finally, in [13] Naylor and the author introduced multisections of 4-manifolds and used them to describe 4manifold operations such as cork twisting and log transforms diagrammatically on a surface.As these descriptions all stem from trisections [7], there are inherent similarities between them. Most obviously, all of the information in these decompositions can be given on a surface. Nevertheless it was not clear how to pass between these descriptions. The first half of this paper is focused on constructing explicit correspondences between certain quotients of the aforementioned sets. These quotients still retain all of the information needed to construct a unique smooth 4-manifold, together with a multisection structure.The second half of the paper is dedicated to showing how any two multi-1

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“…Motivated by work of Castro and Ozbagci [3,6], we want to describe a simple method to paste two compatible -trisections along their boundaries. The main technical observation that makes Theorem 3.1 work is that when we paste two standard pieces Z C 0 ,C 1 along some connected components of C 0 ∪ F all C 1 , the resulting 4-manifold is also a standard piece whenever all the glued surfaces in F all have boundary.…”

Section: Pasting -Trisectionsmentioning

confidence: 99%

“…In Sections 4 and 6, we show that the naive diagrammatic approach to removing neighborhoods of embedded submanifolds yields -trisection diagrams. Additionally, motivated by [6], we prove a pasting lemma in Section 3 which allows us to glue two -trisections along connected boundary components. In Section 6 we use this pasting lemma explicitly to produce a closed trisection diagram for the Cacime Surface.…”

Section: Introductionmentioning

confidence: 99%

Diagrams of *-Trisections

Aranda¹,

Moeller²

2019

Preprint

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In this note we provide a generalization for the definition of a trisection of a 4-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by finding a trisection-theoretic way to perform logarithmic surgery. In addition, we describe how to perform 1-surgery on closed trisections. The insight gained from this description leads us to a classification of the Farey trisection diagrams.

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Trisections of $4$-manifolds via Lefschetz fibrations

Cited by 22 publications

References 0 publications

Trisections of 4-manifolds with boundary

Trisections of 4-manifolds with boundary

Uniqueness of 4-manifolds described as sequences of 3-d handlebodies

Diagrams of *-Trisections

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