Heegaard surfaces and measured laminations, II: Non-Haken 3–manifolds

Li, Tao

doi:10.1090/S0894-0347-06-00520-0

Heegaard surfaces and measured laminations, II: Non-Haken 3–manifolds
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by Tao Li

J. Amer. Math. Soc. 19 (2006), 625-657

DOI: https://doi.org/10.1090/S0894-0347-06-00520-0

Published electronically: February 3, 2006

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Abstract:

A famous example of Casson and Gordon shows that a Haken 3–manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3–manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the generalized Waldhausen conjecture. Another immediate corollary is that for any irreducible non-Haken 3–manifold $M$, there is a number $N$ such that any two Heegaard splittings of $M$ are equivalent after at most $N$ stabilizations.

References

Ian Agol and Tao Li, An algorithm to detect laminar 3-manifolds, Geom. Topol. 7 (2003), 287–309. MR 1988287, DOI 10.2140/gt.2003.7.287
M. Boileau, D. J. Collins, and H. Zieschang, Genus $2$ Heegaard decompositions of small Seifert manifolds, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 1005–1024 (English, with French summary). MR 1150575, DOI 10.5802/aif.1282
Francis Bonahon and Jean-Pierre Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 451–466 (1984) (French). MR 740078, DOI 10.24033/asens.1455
A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275–283. MR 918537, DOI 10.1016/0166-8641(87)90092-7
W. Floyd and U. Oertel, Incompressible surfaces via branched surfaces, Topology 23 (1984), no. 1, 117–125. MR 721458, DOI 10.1016/0040-9383(84)90031-4
David Gabai, Foliations and $3$-manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 609–619. MR 1159248
David Gabai and Ulrich Oertel, Essential laminations in $3$-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. MR 1005607, DOI 10.2307/1971476
Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39–98. MR 0224071
A. E. Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl. 30 (1988), no. 1, 63–88. MR 964063, DOI 10.1016/0166-8641(88)90081-8
Klaus Johannson, Heegaard surfaces in Haken $3$-manifolds, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 91–98. MR 1027902, DOI 10.1090/S0273-0979-1990-15910-5
Klaus Johannson, Topology and combinatorics of 3-manifolds, Lecture Notes in Mathematics, vol. 1599, Springer-Verlag, Berlin, 1995. MR 1439249, DOI 10.1007/BFb0074005
Tsuyoshi Kobayashi, A construction of $3$-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth, Osaka J. Math. 29 (1992), no. 4, 653–674. MR 1192734
Marc Lackenby, The asymptotic behaviour of Heegaard genus, Math. Res. Lett. 11 (2004), no. 2-3, 139–149. MR 2067463, DOI 10.4310/MRL.2004.v11.n2.a1
Tao Li, Laminar branched surfaces in 3-manifolds, Geom. Topol. 6 (2002), 153–194. MR 1914567, DOI 10.2140/gt.2002.6.153
Tao Li, Boundary curves of surfaces with the 4-plane property, Geom. Topol. 6 (2002), 609–647. MR 1941725, DOI 10.2140/gt.2002.6.609

An algorithm to find vertical tori in small Seifert fiber spaces

Heegaard surfaces and measured laminations, I: The Waldhausen conjecture

J. Masters, W. Menasco, and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004), 133–150. MR 2052369
W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44. MR 721450, DOI 10.1016/0040-9383(84)90023-5
John W. Morgan and Peter B. Shalen, Degenerations of hyperbolic structures. II. Measured laminations in $3$-manifolds, Ann. of Math. (2) 127 (1988), no. 2, 403–456. MR 932305, DOI 10.2307/2007061
Yoav Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988), no. 3, 465–481. MR 928492, DOI 10.1007/BF01388781
Yoav Moriah and Jennifer Schultens, Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, Topology 37 (1998), no. 5, 1089–1112. MR 1650355, DOI 10.1016/S0040-9383(97)00072-4

Heegaard splittings of the form H + nK

Ulrich Oertel, Measured laminations in $3$-manifolds, Trans. Amer. Math. Soc. 305 (1988), no. 2, 531–573. MR 924769, DOI 10.1090/S0002-9947-1988-0924769-1

Pretzel Knots

J. H. Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$-dimensional manifolds, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 1–20. MR 1470718, DOI 10.1090/amsip/002.1/01
Hyam Rubinstein and Martin Scharlemann, Comparing Heegaard splittings of non-Haken $3$-manifolds, Topology 35 (1996), no. 4, 1005–1026. MR 1404921, DOI 10.1016/0040-9383(95)00055-0
Martin Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998), no. 1-3, 135–147. MR 1648310, DOI 10.1016/S0166-8641(97)00184-3
E. Sedgwick, An infinite collection of Heegaard splittings that are equivalent after one stabilization, Math. Ann. 308 (1997), no. 1, 65–72. MR 1446199, DOI 10.1007/s002080050064
Michelle Stocking, Almost normal surfaces in $3$-manifolds, Trans. Amer. Math. Soc. 352 (2000), no. 1, 171–207. MR 1491877, DOI 10.1090/S0002-9947-99-02296-5
Friedhelm Waldhausen, Heegaard-Zerlegungen der $3$-Sphäre, Topology 7 (1968), 195–203 (German). MR 227992, DOI 10.1016/0040-9383(68)90027-X
Friedhelm Waldhausen, Some problems on $3$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 313–322. MR 520549
Ying-Qing Wu, Dehn surgery on arborescent knots, J. Differential Geom. 43 (1996), no. 1, 171–197. MR 1424423