Corks with Large Shadow-Complexity and Exotic Four-Manifolds

Naoe, Hironobu

doi:10.1080/10586458.2018.1514332

“…two positive and one negative) additional curls in order to get a quadricolor; however in Fig. 7 Framed links representing the exotic pair W 1 and W 2 (pictures from [30]) this case the resulting graph (K 0 , c) admits a sequence of dipole moves consisting in three 3-dipoles and one 2-dipole (resp. consisting in two 3-dipoles) cancellations yielding a minimal order eight crystallization of S 2 × D 2 (resp.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Example 4 Procedure B, applied to the framed links description given in [30] of an exotic pair (see Fig. 7), allows to obtain two regular 5-colored graphs representing two compact simply-connected PL 4-manifolds W 1 and W 2 with the same topological structure that are not PL-homeomorphic: see Figs.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Kirby diagrams and 5-colored graphs representing compact 4-manifolds

Casali

¹

,

Cristofori

²

2022

Rev Mat Complut

2

0

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It is well-known that in dimension 4 any framed link (L, c) uniquely represents the PL 4-manifold $$M^4(L,c)$$ M 4 ( L , c ) obtained from $${\mathbb {D}}^4$$ D 4 by adding 2-handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram$$(L^{(*)},d)$$ ( L ( ∗ ) , d ) ), the associated PL 4-manifold $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) is obtained from $${\mathbb {D}}^4$$ D 4 by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored graphs: in particular, we describe how to construct algorithmically a (regular) 5-colored graph representing $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) , directly “drawn over” a planar diagram of $$(L^{(*)},d)$$ ( L ( ∗ ) , d ) , or equivalently how to algorithmically obtain a triangulation of $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) . As a consequence, the procedure yields triangulations for any closed (simply-connected) PL 4-manifold admitting handle decompositions without 3-handles. Furthermore, upper bounds for both the invariants gem-complexity and regular genus of $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) are obtained, in terms of the combinatorial properties of the Kirby diagram.

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“…The above construction applied to the framed link descriptions given in [27] of an exotic pair (see Figure 8), allows to obtain two regular 5-colored graphs representing two compact simply-connected PL 4-manifolds W 1 and W 2 with the same topological structure that are not PLhomeomorphic: see Figures 9 and 10.…”

Section: Examplementioning

confidence: 99%

Kirby diagrams and 5-colored graphs representing compact 4-manifolds

Casali,

Cristofori

2021

Preprint

0

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It is well-known that any framed link (L, c) uniquely represents the 3-manifold M 3 (L, c) obtained from S 3 by Dehn surgery along (L, c), as well as the PL 4-manifold M 4 (L, c) obtained from D 4 by adding 2-handles along (L, c), whose boundary coincides with M 3 (L, c). In this paper we study the relationships between the above representation tool in dimension 3 and 4, and the representation theory of compact PL manifolds of arbitrary dimension by edge-coloured graphs: in particular, we describe how to construct a (regular) 5-colored graph representing M 4 (L, c), directly "drawn over" a planar diagram of (L, c). As a consequence, the combinatorial properties of the framed link (L, c) yield upper bounds for both the invariants gem-complexity and (generalized) regular genus of M 4 (L, c).

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“…For example, Martelli studied manifolds (more precisely, 4-dimensional closed smooth manifolds obtained in canonical ways) admitting shadows systematically in [15] and later in [13] with Koda and Naoe. In [17], [18] and [19], Naoe systematically and explicitly studied contractible shadows and 3-dimensional manifolds admitting such shadows. In these studies, there appear various explicit shadows for which we cannot apply Theorem 1.…”

Section: 1mentioning

confidence: 99%

A new explicit way of obtaining special generic maps into the 3-dimensional Euclidean space

Kitazawa

2018

Preprint

0

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A special generic map is a smooth map regarded as a natural generalization of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting special generic maps into the plane are completely determined by Saeki in 1993 and ones admitting such maps into general Euclidean spaces are determined under appropriate conditions.Moreover, if the difference of dimensions of source and target manifolds are not so large, then the diffeomorphism types of source manifolds are often limited; for example, homotopy spheres except standard spheres do not admit special generic maps into Euclidean spaces whose dimensions are not so low. As another example, 4-dimensional simply connected manifolds admitting special generic maps are only manifolds represented as the connected sum of the total spaces of 2-dimensional sphere bundles over the 2-dimensional sphere (or the 4-dimensional standard sphere) and there are many manifolds homeomorphic and not diffeomorphic to these manifolds. These explicit facts make special generic maps attractive objects in the theory of Morse functions and higher dimensional versions and application to algebraic and differentiable topology of manifolds, which is an important study in both singuarity theory of maps and algebraic and differential topology of manifolds.In this paper, we demonstrate a way of construction of special generic maps into the 3-dimensional Euclidean space. For this, first we prepare maps onto 2-dimensional polyhedra regarded as simplicial maps naturally called pseudo quotient maps, which are generalizations of the quotient maps to the spaces of all the connected components of inverse images, so-called Reeb spaces of original smooth maps, being fundamental and important tools in the studies. More precisely, we prepare specific pseudo quotient maps, locally construct maps onto 3-dimensional manifolds, glue them and realize the map as a special generic map into the 3-dimensional space. Most of the procedure for the construction is based on technique the author previously noticed and used in simpler situations.The success of the construction explicitly shows that a class of maps which seems to cover a larger class of source manifolds may not be not so large and that the diffeomorphism types of source manifolds may be restricted as strongly as in the case of special generic maps. We also explain differential topological facts and problems related to this. Introduction.1.1. Backgrounds and fundamental tools. Morse functions and higher dimensional versions and application to algebraic and differentiable topology of manifolds

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Corks with Large Shadow-Complexity and Exotic Four-Manifolds

Cited by 5 publications

References 29 publications

Kirby diagrams and 5-colored graphs representing compact 4-manifolds

Kirby diagrams and 5-colored graphs representing compact 4-manifolds

Kirby diagrams and 5-colored graphs representing compact 4-manifolds

A new explicit way of obtaining special generic maps into the 3-dimensional Euclidean space

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