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
