Bounds for the genus of a normal surface

Jaco, William¹ ; Johnson, Jesse² ; Spreer, Jonathan³ ; Tillmann, Stephan⁴

¹ Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, USA
² Mathematics Department, Oklahoma State University, Stillwater, OK 74078-1058, USA
³ School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
⁴ School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia

Geometry & topology, Tome 20 (2016) no. 3, pp. 1625-1671.

Résumé

This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact orientable 3–manifold in terms of the quadrilaterals in its cell decomposition — different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.

Détail

DOI : 10.2140/gt.2016.20.1625

Classification : 57N10, 57Q15, 57M20, 57N35, 53A05
Keywords: 3–manifold, normal surface, minimal triangulation, efficient triangulation, realisation problem

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