Heegaard Splitting - Classifications

Classifications

There are several classes of three-manifolds where the set of Heegaard splittings is completely known. For example, Waldhausen's Theorem shows that all splittings of are standard. The same holds for lens spaces (as proved by Francis Bonahon and J.P. Otal).

Splittings of Seifert fiber spaces are more subtle. Here, all splittings may be isotoped to be vertical or horizontal (as proved by Yoav Moriah and Jennifer Schultens).

Cooper & Scharlemann (1999) classified splittings of torus bundles (which includes all three-manifolds with Sol geometry). It follows from their work that all torus bundles have a unique splitting of minimal genus. All other splittings of the torus bundle are stabilizations of the minimal genus one.

As of 2008, the only hyperbolic three-manifolds whose Heegaard splittings are classified are two-bridge knot complements, in a paper of Tsuyoshi Kobayashi.