High Quality Content by WIKIPEDIA articles! In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. The importance of Heegaard splittings has grown in recent years as more connections and applications have been found. 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. 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.

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