Heegaard splitting

In the mathematical field of geometric topology, a Heegaard splitting // (listen) is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. In the mathematical field of geometric topology, a Heegaard splitting // (listen) 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. Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with compression bodies. The gluing map is between the positive boundaries of the compression bodies. A closed curve is called essential if it is not homotopic to a point, a puncture, or a boundary component. A Heegaard splitting is reducible if there is an essential simple closed curve α {displaystyle alpha } on H which bounds a disk in both V and in W. A splitting is irreducible if it is not reducible. It follows from Haken's Lemma that in a reducible manifold every splitting is reducible. A Heegaard splitting is stabilized if there are essential simple closed curves α {displaystyle alpha } and β {displaystyle eta } on H where α {displaystyle alpha } bounds a disk in V, β {displaystyle eta } bounds a disk in W, and α {displaystyle alpha } and β {displaystyle eta } intersect exactly once. It follows from Waldhausen's Theorem that every reducible splitting of an irreducible manifold is stabilized.