Minimal triangulations of manifolds

Datta, Basudeb (2007) Minimal triangulations of manifolds Journal of the Indian Institute of Science, 87 (4). pp. 429-449. ISSN 0970-4140

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Finding vertex-minimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertex-minimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present a brief survey on the works done in last 30 years on the following: (i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers n and d, construction of n-vertex triangulations of different d-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices. In Section 1, we have given all the definitions which are required for the remaining part of this article. A reader can start from Section 2 and come back to Section 1 as and when required. In Section 2, we have presented a very brief history of triangulations of manifolds. In Section 3, we have presented examples of several vertex-minimal triangulations. In Section 4, we have presented some interesting results on triangulations of manifolds. In particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem. In Section 5, we have stated several results on minimal triangulations without proofs. Proofs are available in the references mentioned there. We have also presented some open problems/conjectures in Sections 3 and 5.

Item Type:Article
Source:Copyright of this article belongs to The Indian Institute of Science (IISc).
Keywords:Triangulated Manifolds; Combinatorial Manifolds; Minimal Triangulations; Pl Manifolds; Pseudomanifolds; Polytopal Spheres; Stacked Spheres
ID Code:75109
Deposited On:21 Dec 2011 14:11
Last Modified:21 Dec 2011 14:11

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