Reducing Heegaard splittings
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 \(\partial W\).
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