Geodesic
Coverings and minimal triangulations of 3–manifolds
Jaco, William1 ; Rubinstein, J Hyam2 ; Tillmann, Stephan3
1Department of Mathematics, Oklahoma State University, Stillwater OK 74078-1058, USA
2Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia
3School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1257-1265
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This paper uses results on the classification of minimal triangulations of 3–manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space L(4k,2k − 1) and the generalised quaternionic space S3∕Q4k have complexity k, where k ≥ 2. Moreover, it is shown that their minimal triangulations are unique.

DOI : 10.2140/agt.2011.11.1257
Keywords: $3$–manifold, minimal triangulation, layered triangulation, efficient triangulation, complexity, prism manifold, small Seifert fibred space

Jaco, William  1   ; Rubinstein, J Hyam  2   ; Tillmann, Stephan  3

1 Department of Mathematics, Oklahoma State University, Stillwater OK 74078-1058, USA
2 Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia
3 School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia 
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Jaco, William; Rubinstein, J Hyam; Tillmann, Stephan. Coverings and minimal triangulations of 3–manifolds. Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1257-1265. doi: 10.2140/agt.2011.11.1257

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[3] W Jaco, J H Rubinstein, $0$–efficient triangulations of $3$–manifolds, J. Differential Geom. 65 (2003) 61

[4] W Jaco, J H Rubinstein, S Tillmann, Minimal triangulations for an infinite family of lens spaces, J. Topol. 2 (2009) 157

[5] S V Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990) 101

[6] P Orlik, Seifert manifolds, Lecture Notes in Math. 291, Springer (1972)

[7] J H Rubinstein, On $3$–manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979) 129

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