Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds

Asano, Nobutaka

doi:10.48550/arxiv.2010.08239

Cited by 3 publications

(6 citation statements)

References 19 publications

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“…Remark 1. Related to the argument here, Asano [1] considered and solved a more general problem for a class of generic smooth maps into the plane on 4-dimensional closed, connected and orientable manifolds. He considered the class of so-called trisection maps the plane.…”

Section: A Proof Of Main Theorem 1 and Related Problems Results And R...mentioning

confidence: 99%

On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections

Kitazawa¹

2021

Preprint

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Fold maps are fundamental tools in the theory of singularities of differentiable maps and its applications to geometry. They are higher dimensional variants of Morse functions. Classes of special generic maps and round fold maps are important classes of fold maps. Special generic maps are higher dimensional variants of Morse functions on homotopy spheres with exactly two singular points: canonical projections of unit spheres are special generic. Round fold maps are Morse functions obtained as doubles of Morse functions, or fold maps such that the set of all the singular points are embeddings and that the images are concentric. In the present paper, we discuss compositions of these maps with canonical projections. For example, we observe that these compositions for special generic maps of simple classes are regarded as round fold maps in considerable cases. We will also present round fold maps we cannot represent in this way, seeming to be represented so.

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Section: A Proof Of Main Theorem 1 and Related Problems Results And R...mentioning

confidence: 99%

On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections

Kitazawa¹

2021

Preprint

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“…In the second case, let d be a simple closed curve on Σ a 1 obtained from the core of the annulus A on Σ by the surgery from Σ to Σ a 1 . By Lemma 4, the monodromy µ 1 is either t ±1 d or t ±4 d , where t d is the right-handed Dehn twist along d. Next, we introduce 6-tuples of vertical 3-manifolds and a classification of 6-tuples shown in [1]. Let f : X → R 2 be a simplified (2, 0)-trisection.…”

Section: Seementioning

confidence: 99%

“…A 3-manifold obtained as the preimage of an arc on D 2 , as V ij , is called a vertical 3-manifold. In [1] we focused on the 6-tuples…”

Section: Seementioning

confidence: 99%

“…In this case, µ 1 = t ±4 d . From [1], we can assume that the configurations of vanishing cycles are Thus we obtain I(σ 2 V ) = I(V ).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Let µ 1 be the monodromy along a circle between two three-cusped circles of the singular value set, which is an automorphism of Σ a 1 . If µ 1 is the identity map, then the trisection is in some sense "trivial", see [1]. On the other hand, we can find non-trivial replacements of reference paths when µ 1 is not the identity map.…”

Section: Introductionmentioning

confidence: 99%

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Right-left equivalent maps of simplified $(2, 0)$-trisections with different configurations of vanishing cycles

Asano

2021

Preprint

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Trisection maps are certain stable maps from closed 4-manifolds to R 2 . A simpler but reasonable class of trisection maps was introduced by Baykur and Saeki, called a simplified (g, k)-trisection. We focus on the right-left equivalence classes of simplified (2, 0)-trisections. Simplified trisections are determined by their simplified trisection diagrams, which are diagrams on a genus-2 surface consisting of simple closed curves of vanishing cycles with labels. The aim of this paper is to study how the replacement of reference paths changes simplified trisection diagrams up to uppertriangular handle-slides. We show that for a simplified trisection f satisfying a certain condition, there exists at least two simplified (2, 0)-trisections f and f such that f, f and f are right-left equivalent to each other but their simplified trisection diagrams are not related by automorphisms of a genus-2 surface and upper-triangular handle-slides.

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Manifolds admitting special generic maps and their nice generalized multisections

Kitazawa¹

2022

Preprint

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We show that manifolds admitting special generic maps also admit nice generalized multisections.Special generic maps are natural generalized versions of Morse functions with exactly two singular points on closed manifolds, characterizing spheres whose dimensions are not 4 topologically and the 4-dimensional unit sphere, and canonical projections of unit spheres. They are shown to restrict the differentiable structures of spheres etc. and topologies of more general manifolds strongly by Saeki, Sakuma etc., followed by Nishioka, Wrazidlo etc. and followed by the author. Some elementary or important manifolds also admit such maps.(Generalized)multisections of manifolds are nice decompositions of (compact) manifolds, generalizing so-called Heegaard splittings of 3-dimensional manifolds. PL manifolds have been shown to have (generalized) multisections enjoying certain properties by Rubinstein and Tillmann.

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Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds

Cited by 3 publications

References 19 publications

On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections

On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections

Right-left equivalent maps of simplified $(2, 0)$-trisections with different configurations of vanishing cycles

Manifolds admitting special generic maps and their nice generalized multisections

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