Theorems
Alexander's Lemma: Up to isotopy, there is a unique (piecewise linear) embedding of the two-sphere into the three-sphere. (In higher dimensions this is known as the Schoenflies theorem. In dimension two this is the Jordan curve theorem.) This may be restated as follows: the genus zero splitting of is unique.
Waldhausen's Theorem: Every splitting of is obtained by stabilizing the unique splitting of genus zero.
Suppose now that M is a closed orientable three-manifold.
Reidemeister-Singer Theorem: For any pair of splittings and in M there is a third splitting in M which is a stabilization of both.
Haken's Lemma: Suppose that is an essential two-sphere in M and H is a Heegaard splitting. Then there is an essential two-sphere in M meeting H in a single curve.
Read more about this topic: Heegaard Splitting