Heegaard Splitting - Definitions

Definitions

Let V and W be handlebodies of genus g, and let ƒ be an orientation reversing homeomorphism from the boundary of V to the boundary of W. By gluing V to W along ƒ we obtain the compact oriented 3-manifold

Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegard splitting also follows from the work of Smale about handle decompositions from Morse theory.

The decomposition of M into two handlebodies is called a Heegaard splitting, and their common boundary H is called the Heegaard surface of the splitting. Splittings are considered up to isotopy.

The gluing map ƒ need only be specified up to taking a double coset in the mapping class group of H. This connection with the mapping class group was first made by W. B. R. Lickorish.

Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with compression bodies. The gluing map is between the positive boundaries of the compression bodies.

A simple closed curve on a surface is essential if it does not bound a disc on the surface.

A Heegaard splitting is reducible if there is an essential simple closed curve on H which bounds a disk in both V and in W. A splitting is irreducible if it is not reducible. It follows from Haken's Lemma that in a reducible manifold every splitting is reducible.

A Heegaard splitting is stabilized if there are essential simple closed curves and on H where bounds a disk in V, bounds a disk in W, and and intersect exactly once. It follows from Waldhausen's Theorem that every reducible splitting of an irreducible manifold is stabilized.

A Heegaard splitting is weakly reducible if there are disjoint essential simple closed curves and on H where bounds a disk in V and bounds a disk in W. A splitting is strongly irreducible if it is not weakly reducible.

A Heegaard splitting is minimal or minimal genus if there is no other splitting of the ambient three-manifold of lower genus. The minimal value g of the splitting surface is the Heegaard genus of M.