On Heegaard splittings of the sphere \(S^ 3\) (Q1174494)

A Heegaard splitting of genus \(g\) of a 3-manifold \(M\) is a closed orientable surface \(S\) of genus \(g\) which is embedded in \(M\) in such a way that \(M - S\) has two connected components, each of which is homeomorphic to a pretzel of genus \(g\), that is, a connected sum \(\#_ g(S^ 1 \times D^ 2)\). The following theorem is due to F. Waldhausen. Theorem: For every integer \(g\), there is, up to isotopy, a unique Heegaard splitting of genus \(g\) of the 3-sphere. In this paper, the author gives a new proof of this theorem which uses techniques due to Schubert of elimination of singularities of surfaces embedded in the 3-sphere, with the sphere being equipped with a Morse function whose singularities consist of one maximum and one minimum. This proof is elementary (it uses basic differential topology) and is quite different from the proof given by Waldhausen which uses the theorem of Reidemeister-Singer.