Connected sums of unstabilized Heegaard splittings are unstabilized

Bachman, David¹

¹ Pitzer College, 1050 N Mills Ave, Claremont, CA 91711, USA

Geometry & topology, Tome 12 (2008) no. 4, pp. 2327-2378.

Résumé

Let $M_1$ and $M_2$ be closed, orientable 3–manifolds. Let $H_i$ denote a Heegaard surface in $M_i$. We prove that if $H_1\#H_2$ comes from stabilizing a lower genus splitting of $M_1\#M_2$ then one of $H_1$ or $H_2$ comes from stabilizing a lower genus splitting. This answers a question of C Gordon (Problem 3.91 from Kirby’s problem list). We also show that every unstabilized Heegaard splitting has a unique expression as the connected sum of Heegaard splittings of prime 3–manifolds.

Détail

DOI : 10.2140/gt.2008.12.2327

Keywords: Heegaard splitting, connected sum, incompressible surface

Bibliographie

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

[2] J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press (1974)

[3] F Bonahon, J P Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup. $(4)$ 16 (1983)

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

[5] J Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma_{4}=0)$, Lecture Notes in Mathematics 53, Springer (1968)

[6] R Engmann, Nicht-homöomorphe Heegaard–Zerlegungen vom Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume, Abh. Math. Sem. Univ. Hamburg 35 (1970) 33

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

[8] 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

[9] R Kirby, Editor, Problems in low-dimensional topology, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35

[10] H A Masur, Y N Minsky, Geometry of the complex of curves II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902

[11] Y Moriah, J Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998) 1089

[12] R Qiu, Stabilizations of Reducible Heegaard Splittings

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

[14] M Scharlemann, Reconfiguring Qiu's proof of the Gordon Conjecture

[15] M Scharlemann, A Thompson, Thin position for 3–manifolds, from: "Geometric topology (Haifa, 1992)", Contemp. Math. 164, Amer. Math. Soc. (1994) 231

[16] J Schultens, The classification of Heegaard splittings for (compact orientable surface)$ \times S^1$, Proc. London Math. Soc. $(3)$ 67 (1993) 425

[17] J Schultens, The stabilization problem for Heegaard splittings of Seifert fibered spaces, Topology Appl. 73 (1996) 133

[18] F Waldhausen, Heegaard–Zerlegungen der 3–Sphäre, Topology 7 (1968) 195

[19] R Weidmann, Generating tuples of free products, Bull. Lond. Math. Soc. 39 (2007) 393