Heegaard Splitting - Examples

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 \Gamma = S^1 \times \{x_0\} \times \{x_0\} \cup \{x_0\} \times S^1 \times \{x_0\} \cup \{x_0\} \times \{x_0\} \times S^1 . It is an easy exercise to show that V, a regular neighborhood of, is a handlebody as is . Thus the boundary of V in is a Heegaard splitting and this is the standard splitting of . It was proved by Frohman and Hass that any other genus 3 Heegaard splitting of the three-torus is topologically equivalent to this one. Boileau and Otal proved that in general any Heegaard splitting of the three-torus is equivalent to the result of stabilizing this example.

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