Diagrams for Relative Trisections

Castro, Nickolas A.; T., Gay, David; Pinzón-Caicedo, Juanita

doi:10.48550/arxiv.1610.06373

Cited by 2 publications

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“…Next we recall the analogous definitions of relative trisections and their diagrams for 4manifolds with nonempty connected boundary (cf. [12,5]).…”

Section: Gluing Relative Trisectionsmentioning

confidence: 99%

“…It was shown in [5] the induced open book OB on ∂W is given by Σ α , which is the genus p surface with b boundary components obtained from Σ by performing surgery along the α curves. This means that to obtain Σ α , we cut open Σ along each α curve and glue in disks to cap off the resulting boundaries.…”

Section: Gluing Relative Trisectionsmentioning

confidence: 99%

“…Therefore, we get a self-diffeomorphism µ of Σ α fixing ∂Σ α pointwise, which is uniquely determined up to isotopy. Next, we provide some more details (see [5,Theorem 5]) about how to obtain the aforementioned collection of arcs.…”

Section: Gluing Relative Trisectionsmentioning

confidence: 99%

“…Remark 2.6. It is shown in [5] that, up to isotopy, the monodromy of the resulting open book is independent of the choices in the above algorithm.…”

Section: Now We Have Two Ordered Collections Ofmentioning

confidence: 99%

“…Next, we give a very simple version of the general gluing theorem [4] for relatively trisected 4-manifolds. Here we present a different proof -where we use the definition of a relative trisection as given in [5] instead of [12] -for the case of a single boundary component.…”

Section: Now We Have Two Ordered Collections Ofmentioning

confidence: 99%

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Trisections of 4-manifolds via Lefschetz fibrations

Castro¹,

Özbağcı²

2017

Preprint

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We develop a technique for gluing relative trisection diagrams of 4-manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4-manifolds.As an application, we describe a trisection of any closed 4-manifold which admits a Lefschetz fibration over S 2 equipped with a section of square −1, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed 4-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented S 2 -bundle over any closed surface and in particular we draw the corresponding diagrams for T 2 × S 2 and T 2 ×S 2 using our gluing technique. Furthermore, we provide an alternate proof of a recent result of Gay and Kirby which says that every closed 4-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry.

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“…Next we recall the analogous definitions of relative trisections and their diagrams for 4manifolds with nonempty connected boundary (cf. [12,5]).…”

Section: Gluing Relative Trisectionsmentioning

confidence: 99%

Section: Gluing Relative Trisectionsmentioning

confidence: 99%

Section: Gluing Relative Trisectionsmentioning

confidence: 99%

“…Remark 2.6. It is shown in [5] that, up to isotopy, the monodromy of the resulting open book is independent of the choices in the above algorithm.…”

Section: Now We Have Two Ordered Collections Ofmentioning

confidence: 99%

Section: Now We Have Two Ordered Collections Ofmentioning

confidence: 99%

See 3 more Smart Citations

Trisections of 4-manifolds via Lefschetz fibrations

Castro¹,

Özbağcı²

2017

Preprint

Self Cite

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Trisections of 3-Manifolds

Koenig¹

2018

Preprint

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We define a trisection of a closed, orientable three dimensional manifold into three handlebodies, and a notion of stabilization for these trisections. Several examples of trisections are described in detail. We define the trisection genus t(M ) of a 3-manifold, and relate it to the Heegaard genus g(M ), showing that t(M ) ≤ g(M ) ≤ 2t(M ). We show moreover that the bound g(M ) ≤ 2t(M ) is tight. We define stabilizations of trisections and show that all trisections of a 3-manifold are stably equivalent, providing an analogue of the Reidemeister-Singer theorem for trisections. We conclude by showing that there exist complicated trisections of S 3 .

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Diagrams for Relative Trisections

Cited by 2 publications

References 0 publications

Trisections of 4-manifolds via Lefschetz fibrations

Trisections of 4-manifolds via Lefschetz fibrations

Trisections of 3-Manifolds

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