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Triangulations of 3–Manifolds with essential edges
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 5, pp. 1103-1145.

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.

Pour une variété de dimension 3 nous définissons les notions de triangulation essentielle et fortement essentielle. Sous certaines hypothèses sur la topologie ou la géométrie de la variété, nous donnons quatre constructions utilisant différentes méthodes (décompositions de Heegaard, hiérarchies des variétés de Haken, décompositions d’Epstein-Penner et cut-loci des variétés riemanniennes) pour obtenir des triangulations ayant ces propriétés.

Nous montrons aussi qu’une structure de semi-angle est une condition suffisante pour qu’une triangulation d’une 3-variété soit essentielle, et qu’une structure d’angle stricte est une condition suffisante pour qu’une triangulation soit fortement essentielle. De plus, des algorithmes pour tester si une triangulation d’une 3-variété est essentielle ou fortement essentielle sont donnés.

DOI: 10.5802/afst.1477
Craig D. Hodgson 1; J. Hyam Rubinstein 1; Henry Segerman 2; Stephan Tillmann 3

1 Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia
2 Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma, OK 74078, USA
3 School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia
     author = {Craig D. Hodgson and J. Hyam Rubinstein and Henry Segerman and Stephan Tillmann},
     title = {Triangulations of {3{\textendash}Manifolds} with essential edges},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1103--1145},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 24},
     number = {5},
     year = {2015},
     doi = {10.5802/afst.1477},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1477/}
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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 : Mathématiques, Serie 6, Volume 24 (2015) no. 5, pp. 1103-1145. doi : 10.5802/afst.1477. https://afst.centre-mersenne.org/articles/10.5802/afst.1477/

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