quality:corruption
Heegaard splitting

A Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.

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 Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.