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**1 - 2**of**2**### THE INTERSECTING KERNELS OF HEEGAARD SPLITTINGS

"... Let M = V ∪S W be a Heegaard splitting, where V and W are handle bodies of genus g and S is a closed Riemann surface of genus g with ∂V = S and ∂W = S. Consider an essential simple closed curve λ in S. If λ bounds a disk D1 in the manifold V and a disk D2 in W, then by gluing D1 and D2 together alon ..."

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Let M = V ∪S W be a Heegaard splitting, where V and W are handle bodies of genus g and S is a closed Riemann surface of genus g with ∂V = S and ∂W = S. Consider an essential simple closed curve λ in S. If λ bounds a disk D1 in the manifold V and a disk D2 in W, then by gluing D1 and D2 together along λ, we obtain an embedding of the 2-sphere S 2 in the 3-manifold M and so

### SCHARLEMANN)

, 2006

"... Theorem 1 (Waldhausen). Let Σ ⊂ S 3 be a Heegaard surface of genus g> 0. Then Σ is a stabilization of a Heegaard surface of genus g − 1. In [6] J H Rubinstein and M Scharlemann, using Cerf Theory [2], developed tools for comparing Heegaard splittings of irreducible, non-Haken manifolds. As a corolla ..."

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Theorem 1 (Waldhausen). Let Σ ⊂ S 3 be a Heegaard surface of genus g> 0. Then Σ is a stabilization of a Heegaard surface of genus g − 1. In [6] J H Rubinstein and M Scharlemann, using Cerf Theory [2], developed tools for comparing Heegaard splittings of irreducible, non-Haken manifolds. As a corollary of their work they obtained a new proof of Theorem 1. In this note we use Cerf Theory and develop the tools needed for comparing Heegaard splittings of S 3. This allows us to use Rubinstein and Scharlemann’s philosophy and obtain a simpler proof of Theorem 1. We assume familiarity with the basic facts and standard terminology of 3-manifold topology and in particular Heegaard splittings; see [7]. We begin with an outline of the proof. As with many proofs of Theorem 1 we assume the theorem is false and pick Σ to be a minimal genus counterexample; we induct on g, the genus of Σ. A simple application of van Kampen’s theorem shows that if g = 1 then the meridians of the complementary solid tori intersect minimally once and hence Σ is a stabilization of the genus zero splitting of S 3. The heart of the argument (in the following sections) is to show that if g> 1 then Σ weakly reduces. By A Casson and C McA Gordon’s seminal work [1] either Σ reduces or S 3 contains an essential surface. As the latter is impossible, Σ must reduce. Cutting S 3 open along the reducing sphere we obtain 2 balls (say B1 and B2, resp.) and a once punctured surface in each (say S1 and S2, resp.). We attach 3-balls to B1 and B2 and cap off S1 and S2 with disks. It is easy to see that we obtain two Heegaard splittings