- Handlebody
In the mathematical field of

geometric topology , a**handlebody**is a particular kind ofmanifold . Handlebodies are most often used to study3-manifold s, although they can be defined in arbitrarydimension s.**General definition**Let

*G*be a connected finite graph embedded inEuclidean space of dimension n. Let*V*be a closedregular neighborhood of*G*. Then*V*is an n-dimensional handlebody.**3-dimensional handlebodies**Equivalently, a handlebody can be defined as an

orientable 3-manifold-with-boundary containing n pairwise disjoint, properly embedded 2-discs such that the manifold 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.) One can generalize this to higher dimensions also.As a bit of notation, the "genus" of

*V*is the genus of thesurface which forms the boundary of*V*. The graph*G*is called a "spine" of*V*. Finally, it should be noted that, in any fixed genus, there is only one handlebodyup to homeomorphism .The importance of handlebodies in

3-manifold theory comes from their connection withHeegaard splitting s. The importance of handlebodies ingeometric group theory comes from the fact that theirfundamental group is free.A 3-dimensional handlebody is sometimes, particularly in older literature, referred to as a

**cube with handles**.**Examples**Any genus zero handlebody is a three-ball, B

^{3}. A genus one handlebody ishomeomorphic to B^{2}× S^{1}(where S^{1}is thecircle ) and is called a "solidtorus ". All other handlebodies may be obtained by taking the boundaryconnected sum of a collection of solid tori.

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