135 research outputs found
Lower bounds for regular genus and gem-complexity of PL 4-manifolds
Within crystallization theory, two interesting PL invariants for
-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 -manifold , its gem-complexity and its
regular genus satisfy:
where 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.Comment: 17 pages, 3 figures. To appear in Forum Mathematicu
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.Comment: 14 pages, 2 figures; v2: we improved the paper (changes in
Proposition 10; Corollary 9 and Proposition 11 added) taking into account
Theorem 2.6 of arxiv:1310.1991v1 which makes use of our Prop. 6(b)
(arxiv:1309.5728v1). Minor changes have been done, too, in particular to make
references more essentia
An equivalence criterion for PL-manifolds
Aim of the paper is to translate the homeomorphism problem for n-dimensional PL-manifolds, with or without boundary, into an equivalence problem for pseudosimplicial triangulations (i.e. a suitable generalization of simplicial triangulations, where two (curvilinear) simplices may intersect in more than one face), by means of a finite number of moves, called (geometric) dipole moves. Note that the environment of the present work is closely related to the representation method for PL-manifolds via
edge-colorured graphs, since (n+1)-coloured graphs representing n-manifolds are a «discrete way» to visualize suitable pseudosimplicial triangulations. From the graphtheoretical point of view, the equivalence problem was already faced – and solved – in [FG] for closed n-manifolds and in [C2] in the general 3-dimensional setting; here, the
equivalence criterion via (geometric) dipole moves is proved to hold for the whole class of PL n-manifolds; moreover, it is proved to be equivariant with respect to boundary triangulation
Average order of coloured triangulations : the general case
In [Combinatorics of triangulations of 3-manifolds, Trans. Amer. Math. Soc. 337 (2) (1993), 891-906], Luo and Stong introduced the notion of "average edge order" \mu_0(K) = \frac {3 F_0(K)}{E_0(K), K being a triangulation of a closed 3-manifold M with edges and triangles. The present paper extends the above notion to the "average (n-2)-simplex order" of a coloured triangulation K of a compact PL n-manifold with i-simplices: \mu(K) = \frac {n \alpha_{n-1}(K)}{\alpha_{n-2}(K).Main properties of and its relations with the topology of , both in the closed and bounded case, are investigated; the obtained results show the existence of strong analogies with the 3-dimensional simplicial case (see the quoted paper by Luo and Stong, together with [The average edge order of triangulations of 3-manifolds, Osaka J. Math. 33(1986), 761-773] by Tamura)
A Code for m-Bipartite Edge-Coloured Graphs
An (n + 1)-coloured graph is said to
be if m is the maximum integer so that every m-residue
of (i.e. every connected subgraph whose
edges are coloured by only m colours) is bipartite; obviously, every
(n + 1)-coloured graph, with n 2, results to be m-bipartite
for some m, with 2 m n + 1. In this paper, a numerical
of length (2n \textminus{} m + 1) q is assigned
to each m-bipartite (n + 1)-coloured graph of order 2q. Then, it is
proved that,
i.e. if a graph isomorphism exists, which transforms the graphs one
into the other, up to permutation of the edge-colouring. More precisely,
if H is a given group of permutations on the colour set, we face the
problem of algorithmically recognizing H-isomorphic coloured graphs
by means of a suitable defi{}nition of H-code
A catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra
The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 coloured tetrahedra), in [2] (with respect to non-orientable3-manifolds up to 26 coloured tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 coloured tetrahedra): in fact, by automatic generation and analysis of suitable edge-coloured graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting coloured triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to one correspondence with the homeomorphism classes of the represented manifolds
Gem-induced trisections of compact PL -manifolds
The idea of studying trisections of closed smooth -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 -manifolds. The present
paper performs a generalization of these ideas along two different directions:
first, we take in consideration also compact PL -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 -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 for all
-bundles of , and hence it is not finite-to-one.Comment: 25 pages, 14 figures. Updated to most recent versio
Kirby diagrams and 5-colored graphs representing compact 4-manifolds
It is well-known that any framed link uniquely represents the
3-manifold obtained from by Dehn surgery along
, as well as the PL 4-manifold obtained from by
adding 2-handles along , whose boundary coincides with . 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 , directly "drawn
over" a planar diagram of . As a consequence, the combinatorial
properties of the framed link yield upper bounds for both the
invariants gem-complexity and (generalized) regular genus of .Comment: 16 pages, 15 figures. Figures 9 and 10 have been corrected. arXiv
admin note: text overlap with arXiv:1910.0877
TOPOLOGY IN COLORED TENSOR MODELS
From a “geometric topology” point of view, the theory of manifold representation by means of edge-colored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL d-manifold by means of a totally combinatorial tool.
Edge-colored graphs also play an important rôle within colored tensor models theory, considered 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 higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting.
Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the G-degree, have specific relevance in the tensor models framework, show-ing a direct fruitful interaction between tensor models and discrete geometry, via edge-colored graphs.
In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edge-colored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds represented by graphs of a given G-degree. In dimension 4, the goal has already been achieved - via singular 4-manifolds - for all compact PL 4-manifolds with connected boundary up to G-degree 24.
In the same dimension, the existence of colored graphs encoding different PL mani-folds with the same underlying TOP manifold, suggests also to investigate the ability of tensor models to accurately reflect geometric degrees of freedom of Quantum Gravity
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.Comment: 12 pages; accepted for publication in Topology and Its Applications,
volume containing Proceedings of Prague Toposym 201
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