3-pseudomanifolds with preassigned links

Altshuler, Amos

doi:10.1090/s0002-9947-1978-0492298-1

Cited by 10 publications

(4 citation statements)

References 6 publications

(15 reference statements)

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“…First consider the case when there exists a vertex w such that deg L (w) = 5. Let lk L (w)(= lk M (uw)) = C 5 (1,2,3,4,5).…”

Section: Proofsmentioning

confidence: 99%

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Three‐Dimensional Pseudomanifolds on Eight Vertices

Datta¹,

Nilakantan

2008

International Journal of Mathematics and Mathematical Sciences

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A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normal d-pseudomanifolds form a broader class than triangulations of connected closed dmanifolds for d ≥ 3. Here, we classify all the 8-vertex neighbourly normal 3-pseudomanifolds. This gives a classification of all the 8-vertex normal 3-pseudomanifolds. There are 74 such 3-pseudomanifolds, 39 of which triangulate the 3-sphere and other 35 are not combinatorial 3-manifolds. These 35 triangulate six distinct topological spaces. As a preliminary result, we show that any 8-vertex 3-pseudomanifold is equivalent by proper bistellar moves to an 8-vertex neighbourly 3-pseudomanifold. This result is the best possible since there exists a 9-vertex nonneighbourly 3-pseudomanifold which does not allow any proper bistellar moves.

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“…First consider the case when there exists a vertex w such that deg L (w) = 5. Let lk L (w)(= lk M (uw)) = C 5 (1,2,3,4,5).…”

Section: Proofsmentioning

confidence: 99%

“…[13]). In [2], Altshuler has constructed another 8-vertex normal 3-pseudomanifold (namely, N 5 in Example 4). In [14], Lutz has shown that there exist exactly three 8-vertex normal 3-pseudomanifolds which are not combinatorial 3-manifolds (namely, N The topological properties of these normal 3-pseudomanifolds are given in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Three‐Dimensional Pseudomanifolds on Eight Vertices

Datta¹,

Nilakantan

2008

International Journal of Mathematics and Mathematical Sciences

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“…. , j(m)) be a cycle which induces a diagonal-free circuit C in the diagram D of G. If the product S j (1)…”

Section: Complex Reflexion Groupsmentioning

confidence: 99%

“…Before we move on, note that the dual of the polytope 3 T 4 s = {{6, 3} s , {3, 3}} (with s = (1, 1), (2, 0)) is actually a 3-dimensional simplicial complex whose vertex links are isomorphic to the toroidal polyhedron {3, 6} s . A general result due to Altshuler [1] says that, given a finite set of abstract polyhedra which are simplicial 2-complexes, there always exists a finite 3-dimensional simplicial complex whose vertex links belong to the set (with each polyhedron in the set actually occurring as a link). This complex will be an abstract 4-polytope, but will generally not be regular.…”

Section: The Basic Enumeration Techniquementioning

confidence: 99%

Locally unitary groups and regular polytopes

McMullen

Schulte

2002

Advances in Applied Mathematics

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Complex groups generated by involutory reflexions arise naturally in the modern theory of abstract regular polytopes. The paper investigates this relationship, and explains how the enumeration of certain finite universal regular polytopes can be accomplished through the enumeration of certain types of finite complex reflexion groups. In particular, the paper enumerates all the finite groups and their diagrams which arise in this context, and describes the corresponding regular polytopes.

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Classification of Locally Toroidal Regular Polytopes

Schulte

1994

Polytopes: Abstract, Convex and Computational

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In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks 5 and 6. For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.

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3-pseudomanifolds with preassigned links

Cited by 10 publications

References 6 publications

Three‐Dimensional Pseudomanifolds on Eight Vertices

Three‐Dimensional Pseudomanifolds on Eight Vertices

Locally unitary groups and regular polytopes

Classification of Locally Toroidal Regular Polytopes

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