Topological Index Theory for surfaces in 3–manifolds

Bachman, David¹

¹ Department of Mathematics, Pitzer College, Claremont, CA 91711, USA

Geometry & topology, Tome 14 (2010) no. 1, pp. 585-609.

Résumé

The disk complex of a surface in a 3–manifold is used to define its topological index. Surfaces with well-defined topological index are shown to generalize well known classes, such as incompressible, strongly irreducible and critical surfaces. The main result is that one may always isotope a surface $H$ with topological index $n$ to meet an incompressible surface $F$ so that the sum of the indices of the components of $H\setminus N\left(F\right)$ is at most $n$. This theorem and its corollaries generalize many known results about surfaces in 3–manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel’s distance to surfaces with topological index $\ge 2$.

Détail

DOI : 10.2140/gt.2010.14.585

Keywords: Heegaard splitting, minimal surface

Bibliographie

[1] D Bachman, Barriers to topologically minimal surfaces

[2] D Bachman, Heegaard splittings of sufficiently complicated 3–manifolds I: Stabilization

[3] D Bachman, Heegaard splittings of sufficiently complicated 3–manifolds II: Amalgamation

[4] D Bachman, Critical Heegaard surfaces, Trans. Amer. Math. Soc. 354 (2002) 4015

[5] D Bachman, Connected sums of unstabilized Heegaard splittings are unstabilized, Geom. Topol. 12 (2008) 2327

[6] D Bachman, J Johnson, On the existence of high index topologically minimal surfaces, in preparation

[7] D Bachman, S Schleimer, E Sedgwick, Sweepouts of amalgamated 3–manifolds, Algebr. Geom. Topol. 6 (2006) 171

[8] A J Casson, C M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275

[9] M Freedman, J Hass, P Scott, Least area incompressible surfaces in 3–manifolds, Invent. Math. 71 (1983) 609

[10] 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

[11] J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157

[12] J Hempel, 3–manifolds as viewed from the curve complex, Topology 40 (2001) 631

[13] T Kobayashi, Casson–Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3–manifolds, Osaka J. Math. 25 (1988) 553

[14] D Mccullough, Virtually geometrically finite mapping class groups of 3–manifolds, J. Differential Geom. 33 (1991) 1

[15] J Milnor, Morse theory, Annals of Mathematics Studies 51, Princeton University Press (1963)

[16] J T Pitts, J H Rubinstein, Applications of minimax to minimal surfaces and the topology of 3–manifolds, from: "Miniconference on geometry and partial differential equations, 2 (Canberra, 1986)", Proc. Centre Math. Anal. Austral. Nat. Univ. 12, Austral. Nat. Univ. (1987) 137

[17] H Rubinstein, M Scharlemann, Comparing Heegaard splittings of non–Haken 3–manifolds, Topology 35 (1996) 1005

[18] H Rubinstein, M Scharlemann, Transverse Heegaard splittings, Michigan Math. J. 44 (1997) 69

[19] J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3–dimensional manifolds, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1

[20] M Scharlemann, A Thompson, Heegaard splittings of $(\mathrm{surface})\times I$ are standard, Math. Ann. 295 (1993) 549

[21] J Schultens, Additivity of tunnel number for small knots, Comment. Math. Helv. 75 (2000) 353

[22] J Schultens, Heegaard splittings of graph manifolds, Geom. Topol. 8 (2004) 831

[23] M Stocking, Almost normal surfaces in 3–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171