Genus-minimal crystallizations of PL 4-manifolds

Basak, Biplab

doi:10.1007/s13366-017-0334-x

Cited by 10 publications

(10 citation statements)

References 14 publications

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“…As a consequence, we prove that weak semi-simple crystallizations realize the minimum (generalized) regular genus, also in the extended setting of compact 4-manifolds with empty or connected boundary, thus generalizing the analogous result obtained in [3] for the closed case.…”

Section: On the Other Hand If ρsupporting

confidence: 74%

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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: On the Other Hand If ρsupporting

confidence: 74%

“…2 In [5] and [3] two particular types of crystallizations are introduced and studied 4 : they are proved to be "minimal" with respect to regular genus, among all graphs representing the same closed 4-manifold.…”

Section: From Framed Links To 5-colored Graphsmentioning

confidence: 99%

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

Casali¹,

Cristofori²

2019

Preprint

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“…2 cyclic permutations (up to inverse) of ∆ d can be partitioned in (d − 1)! classes, each containing d/2 cyclic permutations,ε (1) ,ε (2) , . .…”

Section: Proof Of the General Resultsmentioning

confidence: 99%

“…where the coefficients F ω G [{t B }] are generating functions of connected bipartite (d + 1)-colored graphs with fixed G-degree ω G . The 1/N expansion of formula (1) describes the rôle of colored graphs (and of their G-degree ω G ) within colored tensor models theory and explains the importance of trying to understand which are the manifolds and pseudomanifolds represented by (d + 1)-colored graphs with a given G-degree.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial properties of the G-degree

Casali

Grasselli

2018

Rev Mat Complut

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A strong interaction is known to exist between edge-colored graphs (which encode PL pseudomanifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the G-degree of the involved graphs, which drives the 1/N expansion in the tensor models context. In the present paper -by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph -we prove that, in any even dimension d ≥ 4, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of (d − 1)!. As a consequence, in even dimension, the terms of the 1/N expansion corresponding to odd powers of 1/N are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations, several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds.

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“…. , ε 4 = 4), we have In [3], we defined weak semi-simple crystallization for closed connected PL 4-manifold. Generalising the definition, here we define weak semi-simple crystallization for a connected compact PL 4-manifold with h boundary components.…”

Section: Lower Bounds For Regular Genus Of Pl 4-manifolds With Boundarymentioning

confidence: 99%

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Basak

Casali

2016

Forum Mathematicum

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Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely gem-complexity and regular genus. In the present paper we prove that, for any closed connected PL 4-manifold M , its gem-complexity k (M ) and its regular genus G(M ) satisfy:where rk(π 1 (M )) = m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of "standard type", involved in existing crystallization catalogues, and their connected sums.MSC 2010 : Primary 57Q15. Secondary 57Q05, 57N13, 05C15.

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Genus-minimal crystallizations of PL 4-manifolds

Cited by 10 publications

References 14 publications

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

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

Combinatorial properties of the G-degree

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

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