An algorithm to detect laminar 3–manifolds

Agol, Ian¹ ; Li, Tao²

¹ Department of Mathematics, University of Illinois at Chicago, 322 SEO m/c 249, 851 S. Morgan Street, Chicago, Illinois 60607-7045, USA
² Department of Mathematics, 401 Math Sciences, Oklahoma State University, Stillwater, Oklahoma 74078-1058, USA

Geometry & topology, Tome 7 (2003) no. 1, pp. 287-309.

Résumé

We show that there are algorithms to determine if a 3–manifold contains an essential lamination or a Reebless foliation.

Détail

DOI : 10.2140/gt.2003.7.287

Keywords: Algorithm, foliation, 3-manifold, lamination

Bibliographie

[1] M Brittenham, Essential laminations in Seifert-fibered spaces, Topology 32 (1993) 61

[2] M Brittenham, Essential laminations and Haken normal form, Pacific J. Math. 168 (1995) 217

[3] M Brittenham, Essential laminations in $I$–bundles, Trans. Amer. Math. Soc. 349 (1997) 1463

[4] M Brittenham, Essential laminations in Seifert-fibered spaces: boundary behavior, Topology Appl. 95 (1999) 47

[5] M Brittenham, R Naimi, R Roberts, Graph manifolds and taut foliations, J. Differential Geom. 45 (1997) 446

[6] D Calegari, Foliations with one-sided branching, Geom. Dedicata 96 (2003) 1

[7] D Calegari, Promoting essential laminations, Invent. Math. 166 (2006) 583

[8] D Calegari, The geometry of $\mathbb{R}$–covered foliations, Geom. Topol. 4 (2000) 457

[9] D Calegari, N M Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003) 149

[10] A Casson, D Jungreis, Convergence groups and Seifert fibered 3–manifolds, Invent. Math. 118 (1994) 441

[11] D Eisenbud, U Hirsch, W Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981) 638

[12] S R Fenley, Laminar free hyperbolic 3–manifolds, Comment. Math. Helv. 82 (2007) 247

[13] W Floyd, U Oertel, Incompressible surfaces via branched surfaces, Topology 23 (1984) 117

[14] D Fried, Fibrations over $S^1$ with pseudo-Anosov monodromy, from: "Travaux de Thurston sur les surfaces", Astérisque 66, Société Mathématique de France (1979) 251

[15] D Gabai, Foliations and the topology of 3–manifolds, J. Differential Geom. 18 (1983) 445

[16] D Gabai, Foliations and the topology of 3–manifolds III, J. Differential Geom. 26 (1987) 479

[17] D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. $(2)$ 136 (1992) 447

[18] D Gabai, Problems in foliations and laminations, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1

[19] D Gabai, Essential laminations and Kneser normal form, J. Differential Geom. 53 (1999) 517

[20] D Gabai, W H Kazez, Group negative curvature for 3–manifolds with genuine laminations, Geom. Topol. 2 (1998) 65

[21] D Gabai, U Oertel, Essential laminations in 3–manifolds, Ann. of Math. $(2)$ 130 (1989) 41

[22] W Haken, Some results on surfaces in 3–manifolds, from: "Studies in Modern Topology", Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.) (1968) 39

[23] W Jaco, personal communication

[24] W Jaco, U Oertel, An algorithm to decide if a 3–manifold is a Haken manifold, Topology 23 (1984) 195

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

[26] W Jaco, J L Tollefson, Algorithms for the complete decomposition of a closed 3–manifold, Illinois J. Math. 39 (1995) 358

[27] M Jankins, W D Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann. 271 (1985) 381

[28] T Li, Laminar branched surfaces in 3–manifolds, Geom. Topol. 6 (2002) 153

[29] J W Morgan, P B Shalen, Degenerations of hyperbolic structures II: Measured laminations in 3–manifolds, Ann. of Math. $(2)$ 127 (1988) 403

[30] R Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv. 69 (1994) 155

[31] S P Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965) 248

[32] U Oertel, Affine laminations and their stretch factors, Pacific J. Math. 182 (1998) 303

[33] R Roberts, J Shareshian, M Stein, Infinitely many hyperbolic 3–manifolds which contain no Reebless foliation, J. Amer. Math. Soc. 16 (2003) 639

[34] J H Rubinstein, An algorithm to recognize the 3–sphere, from: "Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)", Birkhäuser (1995) 601

[35] T Schwider, The classification of essential laminations in Dehn surgeries on the figure-eight knot, PhD thesis, University of Michigan (2001)

[36] A Thompson, Thin position and the recognition problem for $S^3$, Math. Res. Lett. 1 (1994) 613

[37] W P Thurston, A local construction of foliations for three-manifolds, from: "Differential geometry (Proc. Sympos. Pure Math., Vol XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1", Amer. Math. Soc. (1975) 315

[38] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417