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Combinatorial 3-manifolds with few vertices

1974
*
Journal of combinatorial theory. Series A
*

It is proved that every

doi:10.1016/0097-3165(74)90042-9
fatcat:aahl7jtjhrhwdely2yxnelx2bi
*combinatorial**3*-*manifold**with*at most eight*vertices*is a*combinatorial*sphere. ... Thus, the existence of a*3*-*manifold**with*less than 8*vertices*that is not a sphere would lead to the existence of a*3*-*manifold**with*8*vertices*that is not a sphere. ... Every*3*-*manifold**with*at most eight*vertices*is a sphere. This result is best, since the existence of a*3*-*manifold**with*nine*vertices*that is not a sphere was recently shown in [2] . ...##
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The unique 3-neighborly 4-manifold with few vertices

1983
*
Journal of combinatorial theory. Series A
*

This proves the existence of a 3neighborly 4-

doi:10.1016/0097-3165(83)90005-5
fatcat:drl4mlu4ibdflckwn4lwwj5tzi
*manifold**with*nine*vertices*and x =*3*. ... So far as we know this is unsolved. 4-*manifold**with*9 (or 10)*vertices*, the link of each vertex must be a 2-neighborly*3*-sphere*with*eight (or nine)*vertices*. ...##
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Combinatorial manifolds with few vertices

1987
*
Topology
*

WE SHOW that a d-

doi:10.1016/0040-9383(87)90042-5
fatcat:zexf5mdhyrg4hfs22kjag6orsu
*manifold*M*with*less than 3rd/21+*3**vertices*is a sphere and that a ... For d =*3*the assertion follows from the enumerations of all*combinatorial**3*-*manifolds**with*up to 9*vertices*(see Cl]). ... (d. n)-diagram for*combinatorial**manifolds*U. Brehm and W. Kiihnrl*few**vertices*then K, has*few**vertices*, and the last expression can be computed easily in terms of the triangulation. ...##
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Combinatorial 3-Manifolds with Transitive Cyclic Symmetry

2013
*
Discrete & Computational Geometry
*

In addition, we substantially extend the classification of

doi:10.1007/s00454-013-9560-7
fatcat:fhadaggubjcblfhiiktzlqpcfe
*combinatorial**3*-*manifolds**with*transitive cyclic symmetry up to 22*vertices*. ... In this article we give*combinatorial*criteria to decide whether a transitive cyclic*combinatorial*d-*manifold*can be generalized to an infinite family of such complexes, together*with*an explicit construction ... Neighborly*combinatorial**3*-*manifolds**with*dihedral automorphism group*with*up to 19*vertices*as well as cyclic neighborly*combinatorial**3*-*manifolds**with*up to 15*vertices*have already been classified ...##
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Triangulated Manifolds with Few Vertices: Combinatorial Manifolds
[article]

2005
*
arXiv
*
pre-print

In this survey on

arXiv:math/0506372v1
fatcat:amfhlhoszjfftjibhcc5any5oe
*combinatorial*properties of triangulated*manifolds*we discuss various lower bounds on the number of*vertices*of simplicial and*combinatorial**manifolds*. ... 9*vertices*), Altshuler and Steinberg [9] (*combinatorial**3*-*manifolds**with*9*vertices*), Altshuler [2] (neighborly*3*-*manifolds**with*10*vertices*), Lutz [24] (*combinatorial**3*-*manifolds**with*10*vertices*... The numbers of*combinatorial*types of*3*-*manifolds**with*up to 10*vertices*can be found in Table 8 . ...##
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Combinatorial manifolds with complementarity

1994
*
Proceedings of the Indian Academy of sciences. Mathematical sciences
*

We show that if a d-dimensional

doi:10.1007/bf02863418
fatcat:hdzbwwnx2zgzvdndkr6mbfd7bi
*combinatorial**manifold*M*with*n*vertices*satisfies complementarity then d = 0, 2, 4, 8 or 16*with*n = 3d/2 +*3*and IMI is a "*manifold*like a projective plane". ... This classification turned up in the 1987 paper 1"*3*] of Brehm andKiihnel on*combinatorial**manifolds**with**few**vertices*. ... Let M ~ , be a*combinatorial*d-man[fold*with*n*vertices*. If Md~ satisfies complementarity then d=O, 2, 4, 8 or 16*with*n =*3*(d/2)+*3*and IMd, I is a "*manifold*like a projective plane". ...##
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Triangulations with few vertices of manifolds with non-free fundamental group

2019
*
Proceedings of the Royal Society of Edinburgh. Section A Mathematics
*

In particular, we prove that every PL-triangulation of a d-dimensional

doi:10.1017/prm.2018.136
fatcat:nush2u6ki5bljbfvaahyijtiku
*manifold*(d ⩾*3*) whose fundamental group is not free has at least 3d + 1*vertices*. ... As a corollary, every d-dimensional homology sphere that admits a*combinatorial*triangulation*with*less than 3d*vertices*is PL-homeomorphic to Sd. ... This is a difficult question, because there are no standard constructions for triangulations*with**few**vertices*of a given*manifold*, nor there are sufficiently general methods to prove that some specific ...##
###
Triangulations with few vertices of manifolds with non-free fundamental group
[article]

2018
*
arXiv
*
pre-print

In particular, we prove that every PL-triangulation of a d-dimensional

arXiv:1801.05069v1
fatcat:7mv2vly775hcvk55cxvgfmpo7u
*manifold*(d>*3*) whose fundamental group is not free has at least 3d+1*vertices*. ... Another important consequence is that every triangulation*with*small links of M is*combinatorial*. ... This is a difficult question, because there are no standard constructions for triangulations*with**few**vertices*of a given*manifold*, nor there are sufficiently general methods to prove that some specific ...##
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Simplicial Manifolds, Bistellar Flips and a 16-Vertex Triangulation of the Poincaré Homology 3-Sphere

2000
*
Experimental Mathematics
*

We present an algorithm based on bistellar operations that provides a useful tool for the construction of simplicial

doi:10.1080/10586458.2000.10504652
fatcat:2j3seg7n2vfbjhjugknudltvb4
*manifolds**with**few**vertices*. ... As an example, we obtain a 16-vertex triangulation of the Poincar e homology*3*-sphere; we construct an in nite series of non-P L ddimensional spheres*with*d+13*vertices*for d 5; and we show that if a d-*manifold*...*3*)-times the one point Datta-suspension of*3*16 we obtain a non-P L d-sphere*with*d + 13*vertices*. 2 Theorem 7 complements the following two results, which show that triangulated*manifolds**with*\*few*"*vertices*...##
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Combinatorial properties of the K3 surface: Simplicial blowups and slicings
[article]

2010
*
arXiv
*
pre-print

A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from the real projective

arXiv:0909.1453v2
fatcat:hyu4kiz3xnay3knpqdwfrp6nxq
*3*-space onto the 2-sphere*with*the minimum number of*vertices*. ... Starting*with*a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowups. ... Introduction Triangulations of*manifolds**with**few**vertices*have been a growing subject of research during the last years. ...##
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Combinatorial 3-manifolds with transitive cyclic symmetry
[article]

2015
*
arXiv
*
pre-print

In addition, we substantially extend the classification of

arXiv:1112.0940v3
fatcat:j6vhjsl25vg4ze3277lj33ljxa
*combinatorial**3*-*manifolds**with*transitive cyclic symmetry up to 22*vertices*. ... In this article we give*combinatorial*criteria to decide whether a transitive cyclic*combinatorial*d-*manifold*can be generalized to an infinite family of such complexes, together*with*an explicit construction ...*3*-*manifold**with*n = 14 + 4k, k ≥ 0,*vertices*. ...##
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Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes

1995
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Discrete & Computational Geometry
*

We investigate simplicial

doi:10.1007/bf02574049
fatcat:o5soi7ym5vfanpmmokf33wh4ye
*3*-*manifolds*, in particular*3*-spheres,*with**few**vertices*such that the links of all*vertices*are*combinatorially*equivalent (equilinked*3*-spheres), and, simple*3*-*manifolds*, in ... particular*3*-spheres,*with**few*facets such that all facets are*combinatorially*equivalent (equifacetted*3*-spheres). ... Here we define in particular a simplicial equilinked*3*-*manifold*as a simplicial*combinatorial**3*-*manifold**with*the same links. ...##
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Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

2016
*
Israel Journal of Mathematics
*

Here, we give a

doi:10.1007/s11856-016-1330-9
fatcat:5qnikodqmjebldrr3x4fvtwwe4
*combinatorial*criterion on when exactly triangulations of*3*-*manifolds**with*transitive cyclic symmetry can be generalised to an infinite family of such triangulations*with*similarly strong ... In*combinatorial*topology we aim to triangulate*manifolds*such that their topological properties are reflected in the*combinatorial*structure of their description. ... In [26] there is a classification of*combinatorial**3*-*manifolds**with*transitive cyclic symmetry up to 22*vertices*. ...##
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Combinatorial Properties of theK3 Surface: Simplicial Blowups and Slicings

2011
*
Experimental Mathematics
*

A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from the real projective

doi:10.1080/10586458.2011.564546
fatcat:k5ypu3qjnfgszk6jkaqszlizvm
*3*-space onto the 2-sphere*with*the minimum number of*vertices*. ... Starting*with*the minimal 16-vertex triangulation (K 4 ) 16 we resolve its 16 isolated singularities -step by step -by simplicial blowups. ... Introduction Triangulations of*manifolds**with**few**vertices*have been a growing subject of research during the last years. ...##
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Combinatorial 3-Manifolds from Sets of Tetrahedra

2007
*
2007 International Conference on Cyberworlds (CW'07)
*

We propose an algorithm to convert a tetrahedral mesh

doi:10.1109/cw.2007.11
dblp:conf/cw/AtteneFG07
fatcat:efwhm5mc6jg7lhjbpfap7gcolq
*with*singularities to a*combinatorial**3*-*manifold*using only local modifications. ... Furthermore, we demonstrate that the algorithm can be implemented using a flexible state-of-the-art data structure for*manifold*tetrahedral meshes suitable for efficient and general applications. ... More formally, let L be a*combinatorial**3*-ball, and K a homogeneous sub-complex of L*with*no*vertices*in ∂L. ...
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