Volume 12 (2007)

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A proof of Waldhausen's uniqueness of splittings of S³ (after Rubinstein and Scharlemann)

Yo'av Rieck

Geometry & Topology Monographs 12 (2007) 277–284

DOI: 10.2140/gtm.2007.12.277

arXiv: math.GT/0607332

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 S3. In this note we use Cerf Theory and develop the tools needed for comparing Heegaard splittings of S3. 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).

Keywords

Heegaard splittings, 3–sphere, Poincarè conjecture, Cerf theory, the game of Hex

Mathematical Subject Classification

Primary: 57M25, 57M99

References
Publication

Received: 19 October 2005
Revised: 16 July 2006
Accepted: 16 July 2006
Published: 3 December 2007

Authors
Yo'av Rieck
Department of mathematical Sciences
University of Arkansas
Fayetteville
AR 72701
USA