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

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

2006-07-14

arxiv.org/abs/math/0607332v3

Geom. Topol. Monogr. 12 (2007) 277-284

Mathematics

Geometric Topology

This is the version published by Geometry & Topology Monographs on 3 December 2007

Scientific paper

10.2140/gtm.2007.12.277

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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