Trisections of 4-manifolds via Lefschetz fibrations

Castro, Nickolas A.; Özbağcı, Burak

doi:10.48550/arxiv.1705.09854

Cited by 5 publications

(9 citation statements)

References 13 publications

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“…In the last years, the notion of trisection has been extensively studied, also in its relationships with other manifold representation tools, such as handle-decompositions, Lefschetz fibrations, framed links and Kirby diagrams: see, for example, [34], [24], [6], [38], [36] and [25]. In particular, relying on the coincidence between DIFF and PL categories in dimension 4, Bell, Hass, Rubinstein and Tillmann faced the study of unbalanced trisections via (singular) triangulations, by making use of a vertex-labelling by three colors (a tricoloring, which may be a c-tricoloring under suitable connectivity assumptions, or better a ts-tricoloring in case of actually encoding a trisection): see [6] for details.…”

Section: Introductionmentioning

confidence: 99%

Gem-induced trisections of compact PL $4$-manifolds

Casali¹,

Cristofori²

2019

Preprint

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The idea of studying trisections of closed smooth 4-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to colored triangulations associated to the so called simple crystallizations of standard simply-connected 4-manifolds. The present paper performs a generalization of these ideas along two different directions: first, we take in consideration also compact PL 4-manifolds with connected boundary, introducing a possible extension of trisections to the boundary case; then, we analyze the trisections induced not only by simple crystallizations, but by any 5-colored graph encoding a simply-connected 4-manifold. This extended notion is referred to as gem-induced trisection, and gives rise to the G-trisection genus, generalizing the well-known trisection genus. Both in the closed and boundary case, we give conditions on a 5-colored graph which ensure one of its gem-induced trisections -if any -to realize the G-trisection genus, and prove how to determine it directly from the graph itself.Moreover, the existence of gem-induced trisections and an estimation of the G-trisection genus via surgery description is obtained, for each compact simply-connected PL 4-manifold admitting a handle decomposition lacking in 1-handles and 3-handles. As a consequence, we prove that the G-trisection genus equals 1 for all D 2 -bundles of S 2 , and hence it is not finite-to-one.

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Section: Introductionmentioning

confidence: 99%

Gem-induced trisections of compact PL $4$-manifolds

Casali¹,

Cristofori²

2019

Preprint

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“…Here is a list of trisection diagrams for a number of standard (and some exotic) 4-manifolds, along with the corresponding tensor diagram for the trisection bracket. These diagrams are drawn primarily from [15], [7] and [23].…”

Section: Trisection and Tensor Diagrams Of Examplesmentioning

confidence: 99%

4-Manifold Invariants From Hopf Algebras

Chaidez,

Cotler,

Cui

2019

Preprint

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The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperbergs 3manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.

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“…We describe how to obtain trisection diagrams of genus 2g + 2 and g + 2 for D(X g,n ) and D(Y g,n ), respectively (see Example 4.11). It is important to mention that trisection diagrams for such manifolds have been described before in [7] and [5]. But the existance of lower genus diagrams was proven in [2] with no explicit drawings.…”

Section: Introductionmentioning

confidence: 99%

Thin Position through the lens of trisections of 4-manifolds

Aranda

2018

Preprint

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Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to {1, . . . , 1}, and give a relation between the width of M and its double M ∪ id ∂ M . In particular, we describe how to obtain genus 2g + 2 and g + 2 trisection diagrams for sphere bundles over orientable and non-orientable surfaces of genus g, respectively. By last, we study the problem of describing relative handlebodies as cyclic covers of 4-space branched along knotted surfaces from the width perspective.

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Trisections of 4-manifolds via Lefschetz fibrations

Cited by 5 publications

References 13 publications

Gem-induced trisections of compact PL $4$-manifolds

Gem-induced trisections of compact PL $4$-manifolds

4-Manifold Invariants From Hopf Algebras

Thin Position through the lens of trisections of 4-manifolds

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