Enumeration of Non-Orientable 3-Manifolds Using Face-Pairing Graphs and Union-Find

Burton, Benjamin A.

doi:10.1007/s00454-007-1307-x

Cited by 23 publications

(36 citation statements)

References 17 publications

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“…Many examples in dimension 3 can be found in the literature (see Burton [8], Martelli and Petronio [26], Martelli [25] and Matveev [29; 31]), so we turn to higher-dimensional manifolds. A nice spine for CP n can be described by using a technique which was inspired to us by tropical geometry as in Mikhalkin [33].…”

Section: Examplesmentioning

confidence: 99%

“…Tables have been produced in various contexts; see Burton [8], Callahan, Hildebrand and Weeks [9], Frigerio, Martelli and Petronio [14], Martelli [25], Martelli and Petronio [26], Matveev [30; 31] and the references therein (and Table 1 below). Some of these classifications were actually done using the dual viewpoint of singular triangulations, which turns out to be equivalent to Matveev's for the most interesting 3-manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Complexity of PL manifolds

Martelli

2010

Algebr. Geom. Topol.

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We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant).Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with nonpositive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed 4-manifolds of complexity zero (manifolds without 3-handles, doubles of 2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group). 57Q99; 57M99 IntroductionThe complexity c.M / of a compact 3-manifold M (possibly with boundary) was defined in a nice paper of Matveev [29] as the minimum number of vertices of an almost simple spine of M . In that paper he proved the following properties:Finiteness There are finitely many closed irreducible (or cusped hyperbolic) 3-manifolds of bounded complexity.Monotonicity If M S is obtained by cutting M along an incompressible surface S , then c.M S / 6 c.M /.Thanks to the combinatorial nature of spines, it is not hard to classify all manifolds having increasing complexity 0; 1; 2; : : :. Tables have been produced Table 1 below). Some of these classifications were actually done using the dual viewpoint of singular triangulations, which turns out to be equivalent to Matveev's for the most interesting 3-manifolds.We extend here Matveev's complexity from dimension 3 to arbitrary dimension. To do this, we need to choose an appropriate notion of spine. In another paper [27] written in 1973, Matveev defined and studied simple spines of manifolds in arbitrary dimension. A simple spine of a compact manifold is a (locally flat) codimension-1 subpolyhedron with generic singularities, onto which the manifold collapses. If the manifold is closed there cannot be any collapse at all and we therefore need to priorly remove one ball.Simple spines are actually not flexible enough for defining a good complexity. In dimension 3, as an example, any simple spine for S 3 (or, equivalently, D 3 ) is a complicated and unnatural object, such as Bing's house or the abalone. Every simple spine of D 3 has at least one vertex. However, a reasonable complexity must be zero on discs and spheres.To gain more flexibility, Matveev defined in 1988 the more general class of almost simple spines [28; 29] of 3-manifolds. An almost simple polyhedron is a compact polyhedron that can be locally embedded in a simple one. This more general definition allows one to use very natural objects as spines, such as a point for D 3 or a circle for D 2 S 1 : a point is not a vertex by definition, ...

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Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complexity of PL manifolds

Martelli

2010

Algebr. Geom. Topol.

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“…In the closed case (i.e. compact and without boundary), catalogues have already been produced and analysed by many authors [8,19,20], with a particular focus on combinatorial properties of minimal triangulations.On the other hand, the problem of classification in dimension four must take into account that a topological 4-manifold not always admits PL structures or may admit non-equivalent ones. For example, although there exists a classification of simply-connected topological 4-manifolds, long established by Freedman [7], the study of (PL) equivalence classes of such structures, especially with regard to their minimal representatives, is an interesting and still open subject of research.…”

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Generation of Catalogues of PL n-manifolds: Computational Aspects on HPC Systems

Marani¹,

Rivi²,

Cristofori³

2013

SCPE

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Abstract. Within mathematical research, Geometric Topology deals with the study of piecewise-linear n-manifolds, i.e. triangulable spaces which appear locally as the n-dimensional Euclidean space. This paper reports on the computational aspects of an algorithm for generating triangulations of PL 3-and 4-manifolds represented by edge-coloured graphs. As the number of graph vertices is increased the algorithm becomes computationally expensive very quickly, making it a natural candidate for the usage of HPC resources. We present an optimized, parallel version of the algorithm that is suitable for deployment of multi-core systems. Scalability results are discussed on two different platforms, namely an IBM iDataPlex Linux cluster and the IBM supercomputer BlueGene/Q.Key words: High Performance Computing, n-manifolds, coloured triangulations, edge-coloured graphs.AMS subject classifications. 57Q15 -57M15 -68W10.1. Introduction. Geometric Topology deals with piecewise-linear (PL) n-manifolds [2], i.e. compact topological manifolds for which there is a triangulation such that each point has a neighbourhood which is piecewise-linearly isomorphic (PL-homeomorphic) to an affine n-simplex. Catalogues of triangulations of PL n-manifolds are valuable sources of data: they yield examples to test conjectures, calculate invariants and make comparisons; they can offer insight into the structural properties of the represented manifolds and may suggest ideas for further theoretical investigation.In particular, since each compact (topological) 3-manifold admits a PL structure and any two PL structures on the same topological 3-manifold are equivalent (i.e. PL-homeomorphic, see [2]), the study of triangulations of PL 3-manifolds is naturally related to the problem of classification, which is still one of the main topics of 3-dimensional topology. The possibility of representing manifolds by combinatorial structures, together with recent advances in computing power, enabled topologists to construct exhaustive tables of small (i.e. obtained by a small number of simplices) 3-manifolds based on different representation methods. In the closed case (i.e. compact and without boundary), catalogues have already been produced and analysed by many authors [8,19,20], with a particular focus on combinatorial properties of minimal triangulations.On the other hand, the problem of classification in dimension four must take into account that a topological 4-manifold not always admits PL structures or may admit non-equivalent ones. For example, although there exists a classification of simply-connected topological 4-manifolds, long established by Freedman [7], the study of (PL) equivalence classes of such structures, especially with regard to their minimal representatives, is an interesting and still open subject of research. Several examples of different PL 4-manifolds triangulating the same topological 4-manifold have recently been presented and the subject is being continuously updated (see for example [9]), but no exhaustive catalogue is available...

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“…non-orientable) 3-manifolds are available at the Web page http://www.matlas.math.csu.ru/ (resp. in [6,Appendix]). 2 The complexity of a manifold is generally hard to compute from the theoretical point of view, leaving aside the concrete enumeration of its spines.…”

Section: Introductionmentioning

confidence: 99%

A note about complexity of lens spaces

Casali

Cristofori

2014

Forum Mathematicum

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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 ≥ 3, 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 ≥ 3, the following relation: k(L(p, q)) = 5 + 2c(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 .

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Enumeration of Non-Orientable 3-Manifolds Using Face-Pairing Graphs and Union-Find

Cited by 23 publications

References 17 publications

Complexity of PL manifolds

Complexity of PL manifolds

Generation of Catalogues of PL n-manifolds: Computational Aspects on HPC Systems

A note about complexity of lens spaces

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