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Craig D. Hodgson; J. Hyam Rubinstein; Henry Segerman; Stephan Tillmann
Triangulations of 3–Manifolds with essential edges
Annales de la faculté des sciences de Toulouse Sér. 6, 24 no. 5: Numéro Spécial : Actes du Colloque « Topologie et Géométrie de petite dimension », à l’occasion des 60 ans de Michel Boileau, du 24 au 28 juin 2013 à Toulouse (2015), p. 1103-1145, doi: 10.5802/afst.1477
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Résumé - Abstract

We define essential and strongly essential triangulations of $3$–manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3–manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold.

We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3–manifold to be essential, and a strict angle structure is a sufficient condition for a triangulation to be strongly essential. Moreover, algorithms to test whether a triangulation of a 3–manifold is essential or strongly essential are given.

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