Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting 
i// is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
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 Heegaard 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.
...
Wikipedia