of the splitting surface is the Heegaard genus of M. Generalized Heegaard splittings A generalized Heegaard splitting of M is a decomposition into compression bodies and surfaces such that and . The interiors of the compression bodies must be pairwise disjoint and their union must be all of . The surface forms a Heegaard surface for the submanifold of . (Note that here each Vi and Wi is allowed to have more than one component.) A generalized Heegaard splitting is called strongly irreducible if each is strongly irreducible. There is an analogous notion of thin position, defined for knots, for Heegaard splittings. The complexity of a connected surface S, c(S), is defined to be ; the complexity of a disconnected surface is the sum of complexities of its components. The complexity of a generalized Heegaard splitting is the multi-set {c(S_i)}, where the index runs over the Heegaard surfaces in the generalized splitting. These multi-sets can be well-ordered by lexicographical ordering (monotonically decreasing). A generalized Heegaard splitting is thin if its complexity is minimal. Examples Three-sphere The three-sphere is the set of vectors in with length one. Intersecting this with the hyperplane gives a two-sphere. This is the standard genus zero splitting of . Conversely, by Alexander's Trick, all manifolds admitting a genus zero splitting are homeomorphic to . Under the usual identification of with we may view as living in . Then the set of points where each coordinate has norm forms a Clifford torus, . This is the standard genus one splitting of . (See also the discussion at Hopf bundle.) Stabilization Given a Heegaard splitting H in M the stabilization of H is formed by taking the connected sum of the pair with the pair . It is easy to show that the stabilization procedure yields stabilized splittings. Inductively, a splitting is standard if it is the stabilization of a standard splitting. Lens spaces All have a standard splitting of genus one. This is the image of the Clifford torus in under the quotient map used to define the lens space in question. It follows from the structure of the mapping class group of the two-torus that only lens spaces have splittings of genus one. Three-torus Recall that the three-torus is the Cartesian product of three copies of (circles). Let be a point of and consider the graph . It is an easy exercise to show that V,