arXiv subjects

Maria Rita Casali

Publications and source records attributed to Maria Rita Casali.

14 recordsLinked to original sources

arXiv2026

Kirby diagrams, trisections and gems of PL 4-manifolds: relationships, results and open problems

We review the main achievements regarding the interactions between gem theory (which makes use of edge-colored graphs to represent PL-manifolds of arbitrary dimension) and both the classical representation of PL 4-manifolds via Kirby diagrams and the more recent one via trisections. Original results also appear (in particular, about gems representing closed 4-manifolds which need 3-handles in their handle decomposition, as well as about trisection diagrams), together with open problems and further possible applications to the study of compact PL 4-manifolds.

math.GT↗

arXiv2026

Estimating trisection genus via gem theory

Gems are a particular type of edge-colored graphs, dual to colored triangulations, which represent compact PL-manifolds of arbitrary dimension, both in the closed and boundary case. In the present paper, gem theory is used to approach trisections of PL 4-manifolds, so as to prove that: - the graph-defined invariant regular genus is an upper bound for the trisection genus of each closed 4-manifold; - a trisection diagram can be directly obtained from any gem of a closed 4-manifold. Moreover, suitable extensions of the above results are presented for compact 4-manifolds with connected boundary.

math.GT↗

arXiv2024

Trisections of PL 4-manifolds arising from colored triangulations

The purpose of the present paper is twofold: firstly to extend to non-orientable compact 4-manifolds the notion of gem-induced trisection, directly obtained from colored triangulations (or, equivalently, from colored graphs encoding them, called gems); secondly to prove that, both in the orientable and non-orientable case, if the boundary is homeomorphic to a connected sum of sphere bundles over $\mathbb S^1$, gem-induced trisections naturally give rise to trisections of the corresponding closed 4-manifold. As a consequence, an estimation of the trisection genus of any closed orientable 4-manifold in terms of surgery description is obtained via colored triangulations.

math.GT↗

arXiv2022

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

It is well-known that in dimension 4 any framed link $(L,c)$ uniquely represents the PL 4-manifold $M^4(L,c)$ obtained from $\mathbb D^4$ by adding 2-handles along $(L,c)$. Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram $(L^{(*)},d)$), the associated PL 4-manifold $M^4(L^{(*)},d)$ is obtained from $\mathbb D^4$ 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 $M^4(L^{(*)},d)$, directly "drawn over" a planar diagram of $(L^{(*)},d)$, or equivalently how to algorithmically obtain a triangulation of $M^4(L^{(*)},d)$. 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 $M^4(L^{(*)},d)$ are obtained, in terms of the combinatorial properties of the Kirby diagram.

math.GT↗

arXiv2022

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

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 $\mathbb D^2$-bundles of $\mathbb S^2$, and hence it is not finite-to-one.

math.GT↗

arXiv2020

Compact 4-manifolds admitting special handle decompositions

In this paper we study colored triangulations of compact PL 4-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of {\it special handlebody decompositions} for any simply-connected closed PL 4-manifold. In particular, we detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL 4-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariant {\em regular genus, gem-complexity} or {\em G-degree} among all such manifolds with the same second Betti number.

math.GT↗

arXiv2020

Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants

The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called {\it semi-simple} and {\it weak semi-simple crystallizations}, with a particular attention to their properties of minimizing combinatorially defined PL-invariants, such as the {\it regular genus}, the {\it Gurau degree}, the {\it gem-complexity} and the {\it (gem-induced) trisection genus}. The main theorem, yielding a summarizing result on the topic, is an original contribution. Moreover, in the present paper the additivity of regular genus with respect to connected sum is proved to hold for all compact $4$-manifolds with empty or connected boundary which admit weak semi-simple crystallizations.

math.GT↗

arXiv2018

Combinatorial properties of the G-degree

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it G-degree} of the involved graphs, which drives the {\it $1/N$ 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 $d\ge 4$, 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.

math.GT↗

arXiv2018

Topology in colored tensor models via crystallization theory

The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Gurau-degree (or G-degree) of the represented manifolds, in relation with the motivations coming from physics. In fact, the G-degree appears naturally in higher dimensional tensor models as the quantity driving their 1/N expansion, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. In particular, the G-degree of PL-manifolds is proved to be finite-to-one in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PL-manifolds represented by graphs with a fixed G-degree. All these properties have specific relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via crystallization theory.

math-ph↗

arXiv2017

G-degree for singular manifolds

The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-degree 6 is obtained in dimension 3, where all 4-colored graphs represent singular manifolds.

math.GT↗

arXiv2016

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

Within crystallization theory, two interesting PL invariants for $d$-manifolds have been introduced and studied, namely {\it gem-complexity} and {\it regular genus}. In the present paper we prove that, for any closed connected PL $4$-manifold $M$, its gem-complexity $\mathit{k}(M)$ and its regular genus $ \mathcal G(M)$ satisfy: $$\mathit{k}(M) \ \geq \ 3 χ(M) + 10m -6 \ \ \ \text{and} \ \ \ \mathcal G(M) \ \geq \ 2 χ(M) + 5m -4,$$ 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 {\it 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.

math.GT↗

arXiv2013

A note about complexity of lens spaces

Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p,q), with p greater than 2, is bounded by S(p,q)-3, where S(p,q) denotes the sum of all partial quotients in the expansion of q/p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide. Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2 c(L(p,q)), where c(M) denotes the complexity of a 3-manifold M and k(M)+1 is half the minimum order of a crystallization of M.

math.GT↗

arXiv2012

Computing Matveev's complexity via crystallization theory: the boundary case

The notion of Gem-Matveev complexity has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base $\mathbb D^2$ and two exceptional fibers and, therefore, for all torus knot complements.

math.GT↗

arXiv2012

Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams

The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3-manifolds via crystallization theory, yielding the notion of Gem-Matveev complexity; the other one for compact orientable 3-manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the non-orientable case the definition of modified Heegaard complexity and prove that for closed 3-manifolds Gem-Matveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.

math.GT↗