Handlebody


Handlebody

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.

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.

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.

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.


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