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Reducing Heegaard splittings. (English) Zbl 0632.57010
Let $$(W,W')$$ be a Heegaard splitting of a closed 3-manifold M with the Heegaard surface $$F=\partial W=\partial W'$$. If $$(W,W')$$ is not strongly irreducible (i.e. there are essential disks (D,$$\partial D)\subset (W,F)$$, $$(D',\partial D')\subset (W',F)$$ such that $$\partial D\cap \partial D'=\emptyset)$$ then either the splitting is reducible or M contains an incompressible surface. The main tool of the proof is a mild generalization of a result of Haken: if a closed 3-manifold with a given Heegaard splitting contains an essential 2-sphere, then it contains one which meets the Heegaard surface in a single circle. Another application of the Haken result is a sufficient condition for the addition of 2- handles to a 3-manifold to produce an irreducible manifold with incompressible boundary (for another sufficient condition see the reviewer’s paper in Low dimensional topology and Kleinian groups, Symp. Warwick and Durham 1984, Lond. Math. Soc. Lect. Note Ser. 112, 273-285 (1986; Zbl 0621.57005)). Finally the authors answer a question of B. Maskit by showing that if W is a handlebody and the homotopy class of a simple loop in $$\partial W$$ is a proper power in $$\pi_ 1(W)$$, then it is a proper power of an element which is represented by a simple loop in $$\partial W$$.
Reviewer: J.Przytycki

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010)
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##### References:
 [1] Baumslag, G., On generalized free products, Math. Z., 78, 423-438, (1962) · Zbl 0104.24402 [2] Bonahon, F., Cobordism of automorphisms of surfaces, Ann. sci. ec. norm. sup., 16, 4, 237-270, (1983) · Zbl 0535.57016 [3] Bonahon, F.; Otal, J.-P., Scindements de Heegaard des espaces lenticulaires, Ann. sci. ec. norm. sup., 16, 4, 451-466, (1983) · Zbl 0545.57002 [4] Casson, A.J.; Gordon, C.McA., Reducing Heegaard splitting of 3-manifolds, Abstracts amer. math. soc., 4, 2, 182, (1983) [5] A.J. Casson and C.McA. Gordon, Manifolds with irreducible Heegaard spittings of arbitrarily high genus, to appear. [6] M. Domergue and H. Short, Surfaces incompressibles dans les variétés obteneous par chirurgie longitudinale le long d’un noeud de S3, preprint. · Zbl 0607.57010 [7] Frederick, K., Hopfian property of a class of fundamental groups, Comm. pure appl. math., 16, 1-8, (1963) · Zbl 0115.40503 [8] Haken, W., Some results on surfaces in 3-manifolds, (), 34-98, distributed by: Prentice-Hall · Zbl 0194.24902 [9] Hempel, J., 3-manifolds, () · Zbl 0191.22203 [10] Jaco, W., Lectures on three-manifold topology, () · Zbl 0433.57001 [11] Jaco, W., Adding a 2-handle to 3-manifold: an application to property R, Proc. amer. math. soc., 92, 288-292, (1984) · Zbl 0564.57009 [12] K. Johannson, On surfaces in one-relator 3-manifolds, preprint. · Zbl 0624.57012 [13] Karrass, A.; Magnus, W.; Solitar, D., Elements of finite order in a group with a single defining relation, Comm. pure appl. math., 13, 57-66, (1960) · Zbl 0091.02403 [14] Long, D.D., Planar kernels in surface groups, Quart. J. math. Oxford, 35, 305-309, (1984) · Zbl 0556.57006 [15] Maskit, B., Parabolic elements in Kleinian groups, Ann. math., 117, 659-668, (1983) · Zbl 0527.30038 [16] Milnor, J., A unique factorization theorem for 3-manifolds, Amer. J. math., 84, 1-7, (1962) · Zbl 0108.36501 [17] Przytycki, J.H., Incompressibility of surface after Dehn surgery, Michigan math. J., 30, 289-308, (1983) · Zbl 0549.57007 [18] Scharlemann, M., Outermost forks and a theorem of jaco, Proc. rochester confer., (1982) · Zbl 0589.57011 [19] Waldhausen, F., Some problems on 3-manifolds, Proc. symp. pure math., 32, 313-322, (1978)
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