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Combinatorial 3-manifolds with few vertices

Amos Altshuler
1974 Journal of combinatorial theory. Series A  
It is proved that every 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] .  ... 
doi:10.1016/0097-3165(74)90042-9 fatcat:aahl7jtjhrhwdely2yxnelx2bi

The unique 3-neighborly 4-manifold with few vertices

Wolfgang Kühnel, Gunter Lassmann
1983 Journal of combinatorial theory. Series A  
This proves the existence of a 3neighborly 4-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.  ... 
doi:10.1016/0097-3165(83)90005-5 fatcat:drl4mlu4ibdflckwn4lwwj5tzi

Combinatorial manifolds with few vertices

U. Brehm, W. Kühnel
1987 Topology  
WE SHOW that a d-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.  ... 
doi:10.1016/0040-9383(87)90042-5 fatcat:zexf5mdhyrg4hfs22kjag6orsu

Combinatorial 3-Manifolds with Transitive Cyclic Symmetry

Jonathan Spreer
2013 Discrete & Computational Geometry  
In addition, we substantially extend the classification of 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  ... 
doi:10.1007/s00454-013-9560-7 fatcat:fhadaggubjcblfhiiktzlqpcfe

Triangulated Manifolds with Few Vertices: Combinatorial Manifolds [article]

Frank H. Lutz
2005 arXiv   pre-print
In this survey on 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 .  ... 
arXiv:math/0506372v1 fatcat:amfhlhoszjfftjibhcc5any5oe

Combinatorial manifolds with complementarity

Basudeb Datta
1994 Proceedings of the Indian Academy of sciences. Mathematical sciences  
We show that if a d-dimensional 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".  ... 
doi:10.1007/bf02863418 fatcat:hdzbwwnx2zgzvdndkr6mbfd7bi

Triangulations with few vertices of manifolds with non-free fundamental group

Petar Pavešić
2019 Proceedings of the Royal Society of Edinburgh. Section A Mathematics  
In particular, we prove that every PL-triangulation of a d-dimensional 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  ... 
doi:10.1017/prm.2018.136 fatcat:nush2u6ki5bljbfvaahyijtiku

Triangulations with few vertices of manifolds with non-free fundamental group [article]

Petar Pavešić
2018 arXiv   pre-print
In particular, we prove that every PL-triangulation of a d-dimensional 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  ... 
arXiv:1801.05069v1 fatcat:7mv2vly775hcvk55cxvgfmpo7u

Simplicial Manifolds, Bistellar Flips and a 16-Vertex Triangulation of the Poincaré Homology 3-Sphere

Anders Björner, Frank H. Lutz
2000 Experimental Mathematics  
We present an algorithm based on bistellar operations that provides a useful tool for the construction of simplicial 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  ... 
doi:10.1080/10586458.2000.10504652 fatcat:2j3seg7n2vfbjhjugknudltvb4

Combinatorial properties of the K3 surface: Simplicial blowups and slicings [article]

Jonathan Spreer, Wolfgang Kühnel
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 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.  ... 
arXiv:0909.1453v2 fatcat:hyu4kiz3xnay3knpqdwfrp6nxq

Combinatorial 3-manifolds with transitive cyclic symmetry [article]

Jonathan Spreer
2015 arXiv   pre-print
In addition, we substantially extend the classification of 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.  ... 
arXiv:1112.0940v3 fatcat:j6vhjsl25vg4ze3277lj33ljxa

Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes

J. Bokowski, P. Schuchert
1995 Discrete & Computational Geometry  
We investigate simplicial 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.  ... 
doi:10.1007/bf02574049 fatcat:o5soi7ym5vfanpmmokf33wh4ye

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

Benjamin Burton, Jonathan Spreer
2016 Israel Journal of Mathematics  
Here, we give a 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.  ... 
doi:10.1007/s11856-016-1330-9 fatcat:5qnikodqmjebldrr3x4fvtwwe4

Combinatorial Properties of theK3 Surface: Simplicial Blowups and Slicings

Jonathan Spreer, Wolfgang Kühnel
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 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.  ... 
doi:10.1080/10586458.2011.564546 fatcat:k5ypu3qjnfgszk6jkaqszlizvm

Combinatorial 3-Manifolds from Sets of Tetrahedra

Marco Attene, Massimo Ferri, Daniela Giorgi
2007 2007 International Conference on Cyberworlds (CW'07)  
We propose an algorithm to convert a tetrahedral mesh 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.  ... 
doi:10.1109/cw.2007.11 dblp:conf/cw/AtteneFG07 fatcat:efwhm5mc6jg7lhjbpfap7gcolq
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