resulting from cutting along the discs is a 3-ball. It's instructive to imagine how to reverse this process to get a handlebody. (Sometimes the orientability hypothesis is dropped from this last definition, and one gets a more general kind of handlebody with a non-orientable handle.) The genus of a handlebody is the genus of its boundary surface. Up to homeomorphism, there is exactly one handlebody of any non-negative integer genus. The importance of handlebodies in 3-manifold theory comes from their connection with Heegaard splittings. The importance of handlebodies in geometric group theory comes from the fact that their fundamental group is free. A 3-dimensional handlebody is sometimes, particularly in older literature, referred to as a cube with handles. Examples Let G be a connected finite graph embedded in Euclidean space of dimension n. Let V be a closed regular neighborhood of G in the Euclidean space. Then V is an n-dimensional handlebody. The graph G is called a spine of V. Any genus zero handlebody is homeomorphic to the three-ball B3. A genus one handlebody is homeomorphic to B2 × S1 (where S1 is the circle) and is called a solid torus. All other handlebodies may be obtained by taking the boundary-connected sum of a collection of solid tori. See also Handle decomposition References Geometric topology Surgery theory