logo AFST
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.

Published online:
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
@article{AFST_2015_6_24_5_1103_0,
     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, 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/}
}
TY  - JOUR
TI  - Triangulations of 3–Manifolds with essential edges
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2015
DA  - 2015///
SP  - 1103
EP  - 1145
VL  - Ser. 6, 24
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1477/
UR  - https://doi.org/10.5802/afst.1477
DO  - 10.5802/afst.1477
LA  - en
ID  - AFST_2015_6_24_5_1103_0
ER  - 
%0 Journal Article
%T Triangulations of 3–Manifolds with essential edges
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2015
%P 1103-1145
%V Ser. 6, 24
%N 5
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1477
%R 10.5802/afst.1477
%G en
%F AFST_2015_6_24_5_1103_0
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/

[1] Anisov (S.).— Cut loci in lens manifolds. C. R. Math. Acad. Sci. Paris 342, no. 8, p. 595-600 (2006). | MR: 2217922 | Zbl: 1103.53016

[2] Aschenbrenner (M.), Friedl (S.) and Wilton (H.).— Decision problems for 3-manifolds and their fundamental groups, arXiv:1405.6274 (2014).

[3] Burton (B. A.), Budney (R.), Pettersson (W.) et al.— Regina: Software for 3-manifold topology and normal surface theory, http://regina.sourceforge.net/, (1999-2012).

[4] Culler (M.), Dunfield (N. M.), and Weeks (J. R.).— SnapPy, a computer program for studying the geometry and topology of 3-manifolds, .

[5] Dyson (V. H.).— The word problem and residually finite groups, Notices Amer. Math. Soc. 11, p. 734 (1964).

[6] Epstein (D. B. A.).— Projective planes in 3-manifolds, Proc. London Math. Soc. (2) 76, p. 180-184 (1962). | MR: 140121 | Zbl: 0107.40304

[7] Epstein (D. B. A.) and Penner (R. C.).— Euclidean decompositions of noncompact hyperbolic manifolds, Journal of Differential Geometry 27, no. 1, p. 67-80 (1988). | MR: 918457 | Zbl: 0611.53036

[8] Ehrlich (P.) and Im Hof (H.-C.).— Dirichlet regions in manifolds without conjugate points, Commentarii Math. Helv. 54, p. 642-658 (1979). | MR: 552682 | Zbl: 0419.53025

[9] Friedl (S.) and Wilton (H.).— The membership problem for 3-manifold groups is solvable, arXiv:1401.2648, 2014, to appear in Algebraic & Geometric Topology.

[10] Gelfand (I. M.), Kapranov (M. M.), and Zelevinsky (A. V.).— Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA (1994). | MR: 1264417 | Zbl: 1138.14001

[11] Garoufalidis (S.), Hodgson (C. D.), Rubinstein (J. H.) and Segerman (H.).— 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, Geometry & Topology 19, p. 2619-2689 (2015). | MR: 3416111

[12] Guéritaud (F.).— On canonical triangulations of once-punctured torus bundles and two-bridge link complements, with an appendix by David Futer, Geometry & Topology 10, p. 1239-1284 (2006). | MR: 2255497 | Zbl: 1130.57024

[13] Guéritaud (F.).— Delaunay triangulations of lens spaces, arXiv:0901.2738v2, (2009).

[14] Hatcher (A.).— Notes on Basic 3-manifold Topology, available from .

[15] Hempel (J.).— Residual finiteness for 3-manifolds, in: S. M. Gersten and John R. Stallings (eds.), Combinatorial Group Theory and Topology, pp. 379-396, Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ (1987). | MR: 895623 | Zbl: 0772.57002

[16] Hodgson (C. D.), Rubinstein (J. H.) and Segerman (H.).— Triangulations of hyperbolic 3-manifolds admitting strict angle structures, Journal of Topology. 5, p. 887-908 (2012). | MR: 3001314 | Zbl: 1262.57018

[17] Jaco (W.) and Rubinstein (J. H.).— 0-efficient triangulations of 3-manifolds, J. Differential Geometry 65, p. 61-168 (2003). | MR: 2057531 | Zbl: 1068.57023

[18] Jaco (W.) and Rubinstein (J. H.).— Layered-triangulations of 3-manifolds, arXiv:math/0603601. | MR: 2866933

[19] Jaco (W.), Rubinstein (J. H.) and Tillmann (S.).— Minimal triangulations for an infinite family of lens spaces, Journal of Topology 2 p. 157-180 (2009). | MR: 2499441 | Zbl: 1227.57026

[20] Jaco (W.) and Tollefson (J. L.).— Algorithms for the complete decomposition of a closed 3-manifold, Illinois J. Math. 39, no. 3, p. 358-406 (1995). | MR: 1339832 | Zbl: 0858.57018

[21] Kang (E.) and Rubinstein (J. H.).— Ideal triangulations of 3-manifolds II; taut and angle structures, Algebraic & Geometric Topology 5, p. 1505-1533 (2005). | MR: 2186107 | Zbl: 1096.57018

[22] Lackenby (M.).— Taut ideal triangulations of 3-manifolds, Geom. Topol. 4, p. 369-395 (2000). | MR: 1790190 | Zbl: 0958.57019

[23] Luo (F.) and Tillmann (S.).— Angle structures and normal surfaces, Trans. Amer. Math. Soc. 360, no. 6, p. 2849-2866 (2008). | MR: 2379778 | Zbl: 1152.57004

[24] Matveev (S.).— Algorithmic topology and classification of 3-manifolds. Algorithms and Computation in Mathematics, 9. Springer, Berlin (2003). | MR: 1997069 | Zbl: 1048.57001

[25] Mostowski (A. W.).— On the decidability of some problems in special classes of groups, Fund. Math. 59, p. 123-135 (1966). | MR: 224693 | Zbl: 0143.03701

[26] Niblo (G. A.).— Separability properties of free groups and surface groups, J. Pure Appl. Algebra 78, p. 77-84 (1992). | MR: 1154898 | Zbl: 0766.57002

[27] Ramanayake (S.).— 0-efficient triangulations of Haken three-manifolds, Ph.D. thesis, submitted, University of Melbourne (2015).

[28] Ratcliffe (J.).— Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, Springer, Second Edition (2006). | MR: 2249478 | Zbl: 1106.51009

[29] Rubinstein (J. H.).— One-sided Heegaard splittings of 3-manifolds, Pacific J. of Math. 76, p. 185-200 (1978). | MR: 488064 | Zbl: 0394.57013

[30] Schleimer (S.).— Almost normal Heegaard splittings, Ph.D. Thesis, University of California, Berkeley. 2001. 81 pp. Available from http://homepages.warwick.ac.uk/masgar/index.html. | MR: 2702296

[31] Scott (P.).— Subgroups of surface groups are almost geometric, J. London Math. Soc. 17, p. 555-565 (1978). | MR: 494062 | Zbl: 0412.57006

[32] Segerman (H.) and Tillmann (S.).— Pseudo-developing maps for ideal triangulations I: Essential edges and generalised hyperbolic gluing equations, Topology and Geometry in Dimension Three: Triangulations, Invariants, and Geometric Structures (Proceedings of the Jacofest conference), AMS Contemporary Mathematics 560, p. 85-102 (2011). | MR: 2866925

[33] Seifert (H.) and Threlfall (W.).— Lehrbuch der Topologie, AMS Chelsea Publishing, 31, (1934) (reprinted 2003). | Zbl: 0009.08601

[34] Stallings (J. R.).— On fibering certain 3-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice Hall, p. 95-100 | MR: 158375 | Zbl: 1246.57049

[35] Stocking (M.).— Almost normal surfaces in 3-manifolds, Trans. Amer. Math. Soc. 352, no. 1, p. 171-207 (2000). | MR: 1491877 | Zbl: 0933.57016

[36] Thompson (A.).— Thin position and the recognition problem for S 3 , Math. Res. Lett. 1, no. 5, p. 613-630 (1994). | MR: 1295555 | Zbl: 0849.57009

[37] Waldhausen (F.).— Heegaard-Zerlegungen der 3-Sphäre, Topology 7 no. 2, p. 195-203 (1968). | MR: 227992 | Zbl: 0157.54501

[38] Waldhausen (F.).— On irreducible 3-manifolds which are sufficiently large, Annals of Math. 87, p. 56-88 (1968). | MR: 224099 | Zbl: 0157.30603

Cited by Sources: