A proof of Waldhausen's uniqueness of splittings of S^3 (after Rubinstein and Scharlemann)

Yo'av Rieck

doi:10.2140/gtm.2007.12.277

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A proof of Waldhausen's uniqueness of splittings of S^3 (after Rubinstein and Scharlemann)

Yo'av Rieck

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Abstract

In [Topology 35 (1996) 1005--1023] J H Rubinstein and M Scharlemann, using Cerf Theory, developed tools for comparing Heegaard splittings of irreducible, non-Haken manifolds. As a corollary of their work they obtained a new proof of Waldhausen's uniqueness of Heegaard splittings of S^3. 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 Waldhausen's Theorem. The combinatorics we use are very similar to the game Hex and requires that Hex has a winner. The paper includes a proof of that fact (Proposition 3.6).

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Yo'av Rieck. 2009-03-31. A proof of Waldhausen's uniqueness of splittings of S^3 (after Rubinstein and Scharlemann). https://doi.org/10.2140/gtm.2007.12.277

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