Handlebody

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In the mathematical field of geometric topology, a handlebody is a particular kind of manifold. Handlebodies are most often used to study 3-manifolds, although they can be defined in arbitrary dimensions.

[edit] General definition

Let G be a connected finite graph embedded in Euclidean space of dimension n. Let V be a closed regular neighborhood of G. Then V is an n-dimensional handlebody.

[edit] 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 the surface 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 handlebody up to homeomorphism.

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.

[edit] Examples

Any genus zero handlebody is a 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.