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

Basak, Biplab; Casali, Maria Rita

doi:10.1515/forum-2015-0080

Cited by 15 publications

(35 citation statements)

References 30 publications

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“…which is used by many authors within tensor models theory (see for example [15]). 2 Actually, we are able to prove that, if d ≥ 4 is even, under rather weak hypotheses, the G-degree is multiple of (d-1)! (or, equivalently, the reduced G-degree is even).…”

Section: Proof Of the General Resultsmentioning

confidence: 92%

“…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%

“…Note that, in this case, the permutationε such that ρ(Γ) = ρε(Γ) coincides with the permutation ε of point (a); moreover, ρ(Γ) = G(M 4 ) holds, in virtue of the inequality G(M 4 ) ≥ 2χ(M 4 ) + 5m − 4, proved in[2] 6. ε turns out to be the permutation of ∆ 4 associated to ε = (0, 1, 2, 3, 4).…”

mentioning

confidence: 86%

“…According to [2], for each order 2p crystallization (Γ, γ) of a closed PL 4-manifold M 4 , with rk(π 1 (M 4 )) = m0, let us set g j,k,l = 1 + m + t j,k,l , with t j,k,l ∀j, k, l ∈ ∆ 4 .…”

Section: Moreovermentioning

confidence: 99%

“…,ε (d) say, so thatd i=1 j∈Z d+1 gε(i) j ,ε (i) j+1 = 2 r,s∈∆ d g r,s . 3In this way, one can reobtain -for d ≥ 4 even -relation(2).…”

mentioning

confidence: 99%

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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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Section: Proof Of the General Resultsmentioning

confidence: 92%

“…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%

mentioning

confidence: 86%

Section: Moreovermentioning

confidence: 99%

“…,ε (d) say, so thatd i=1 j∈Z d+1 gε(i) j ,ε (i) j+1 = 2 r,s∈∆ d g r,s . 3In this way, one can reobtain -for d ≥ 4 even -relation(2).…”

mentioning

confidence: 99%

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Combinatorial properties of the G-degree

Casali

Grasselli

2018

Rev Mat Complut

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Handle decompositions for a class of closed orientable PL 4-manifolds

Basak

Binjola

2023

Indian J Pure Appl Math

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Genus-minimal crystallizations of PL 4-manifolds

Basak

2017

Beitr Algebra Geom

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For d ≥ 2, the regular genus of a closed connected PL d-manifold M is the least genus (resp., half of genus) of an orientable (resp., a non-orientable) surface into which a crystallization of M imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every 3-manifold equals its Heegaard genus. For every closed connected PL 4-manifold M , it is known that its regular genus G(M ) is at least 2χ(M ) + 5m − 4, where m is the rank of the fundamental group of M . In this article, we introduce the concept of "weak semi-simple crystallization" for every closed connected PL 4-manifold M , and prove that G(M ) = 2χ(M ) + 5m − 4 if and only if M admits a weak semi-simple crystallization. We then show that the PL invariant regular genus is additive under the connected sum within the class of all PL 4-manifolds admitting a weak semi-simple crystallization. Also, we note that this property is related to the 4-dimensional Smooth Poincaré Conjecture.MSC 2010 : Primary 57Q15. Secondary 57Q05; 57N13; 05C15.

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

Cited by 15 publications

References 30 publications

Combinatorial properties of the G-degree

Combinatorial properties of the G-degree

Handle decompositions for a class of closed orientable PL 4-manifolds

Genus-minimal crystallizations of PL 4-manifolds

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