Attaching handlebodies to 3–manifolds

Lackenby, Marc¹

¹ Mathematical Institute, Oxford University, 24–29 St Giles’, Oxford, OX1 3LB, United Kingdom

Geometry & topology, Tome 6 (2002) no. 2, pp. 889-904.

Résumé

The main theorem of this paper is a generalisation of well known results about Dehn surgery to the case of attaching handlebodies to a simple 3–manifold. The existence of a finite set of ‘exceptional’ curves on the boundary of the 3–manifold is established. Provided none of these curves is attached to the boundary of a disc in a handlebody, the resulting manifold is shown to be word hyperbolic and ‘hyperbolike’. We then give constructions of gluing maps satisfying this condition. These take the form of an arbitrary gluing map composed with powers of a suitable homeomorphism of the boundary of the handlebodies.

Détail

DOI : 10.2140/gt.2002.6.889

Keywords: 3–manifold, handlebody, word hyperbolic

Bibliographie

[1] A Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994) 207

[2] S A Bleiler, C D Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996) 809

[3] , Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284

[4] S Gallot, D Hulin, J Lafontaine, Riemannian geometry, Universitext, Springer (2004)

[5] J Hass, Minimal surfaces in manifolds with $S^1$ actions and the simple loop conjecture for Seifert fibered spaces, Proc. Amer. Math. Soc. 99 (1987) 383

[6] J Hass, J H Rubinstein, S Wang, Boundary slopes of immersed surfaces in 3–manifolds, J. Differential Geom. 52 (1999) 303

[7] J Hass, S Wang, Q Zhou, On finiteness of the number of boundary slopes of immersed surfaces in 3–manifolds, Proc. Amer. Math. Soc. 130 (2002) 1851

[8] C D Hodgson, S P Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Ann. of Math. $(2)$ 162 (2005) 367

[9] M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243

[10] H Masur, Measured foliations and handlebodies, Ergodic Theory Dynam. Systems 6 (1986) 99

[11] J W Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, from: "The Smith conjecture (New York, 1979)", Pure Appl. Math. 112, Academic Press (1984) 37

[12] M Scharlemann, Y Q Wu, Hyperbolic manifolds and degenerating handle additions, J. Austral. Math. Soc. Ser. A 55 (1993) 72

[13] W P Thurston, The Geometry and Topology of 3–manifolds, Princeton University (1980)

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