Genus two Heegaard splittings

Hass, Joel

doi:10.1090/S0002-9939-1992-1070519-7

Genus two Heegaard splittings
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Proc. Amer. Math. Soc. 114 (1992), 565-570 Request permission

Abstract:

It is shown that a $3$-manifold has a finite number of genus two Heegaard splittings. A corollary is that the mapping class group is finite if the manifold is non-Haken.

References

Michael T. Anderson, Curvature estimates for minimal surfaces in $3$-manifolds, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 1, 89–105. MR 803196
Francis Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983), no. 3, 305–314 (French). MR 710104, DOI 10.1016/0040-9383(83)90016-2
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
Michel Boileau and Jean-Pierre Otal, Groupe des difféotopies de certaines variétés de Seifert, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 1, 19–22 (French, with English summary). MR 849619
Hyeong In Choi and Richard Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985), no. 3, 387–394. MR 807063, DOI 10.1007/BF01388577
John L. Friedman and Donald M. Witt, Homotopy is not isotopy for homeomorphisms of $3$-manifolds, Topology 25 (1986), no. 1, 35–44. MR 836722, DOI 10.1016/0040-9383(86)90003-0
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
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497

Homotopy and isotopy of

-manifolds

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
J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR 142125, DOI 10.2307/2372800
Jon T. Pitts and J. H. Rubinstein, Existence of minimal surfaces of bounded topological type in three-manifolds, Miniconference on geometry and partial differential equations (Canberra, 1985) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 10, Austral. Nat. Univ., Canberra, 1986, pp. 163–176. MR 857665
Jon T. Pitts and J. H. Rubinstein, Equivariant minimax and minimal surfaces in geometric three-manifolds, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 303–309. MR 940493, DOI 10.1090/S0273-0979-1988-15652-2

Manifolds with infinitely many non-isotopic Heegaard splittings of minimal genus

Peter Scott, There are no fake Seifert fibre spaces with infinite $\pi _{1}$, Ann. of Math. (2) 117 (1983), no. 1, 35–70. MR 683801, DOI 10.2307/2006970

The geometry and topology of

-manifolds

An orbifold theorem for

-manifolds

Minimal embeddings of surfaces in hyperbolic

-manifolds

Friedhelm Waldhausen, Recent results on sufficiently large $3$-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. 21–38. MR 520520
Brian White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200. MR 1101226, DOI 10.1512/iumj.1991.40.40008