Datta, Basudeb (2007) Minimal Triangulations of Manifolds. In: Journal of the Indian Institute of Science, 87 (4).

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Abstract
Finding vertexminimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertexminimal 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 nvertex triangulations of different ddimensional 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 vertexminimal 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:  Journal Article 

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Additional Information:  Copyright of this article belongs to The Indian Institute of Science (IISc). 
Department/Centre:  Division of Physical & Mathematical Sciences > Mathematics 
Date Deposited:  03 Feb 2012 06:05 
Last Modified:  03 Feb 2012 06:05 
URI:  http://eprints.iisc.ernet.in/id/eprint/43365 
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