Heegaard splitting
In mathematics, in the sub-field of geometric topology, a Heegaard splitting
is a method for presenting three-manifolds.
To be precise, suppose that V and W are handlebodies of the same genus. Choose now a homeomorphism f from the boundary of V to the boundary of W. Now form the quotient space
This is clearly a three-manifold. The remarkable fact is that every three-manifold admits such a presentation. 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.
Note also that the gluing map f need only be specified up to taking a double coset in the mapping class group of the boundary of V. This connection with the mapping class group was first made by W. R. Lickorish.