Handlebody
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In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Handles are used to particularly study 3-manifolds.
[edit] n-dimensional handlebodies
If 
is a n-dimensional manifold with boundary, and 
is an embedding, the n-dimensional manifold with boundary
is said to be obtained from by attaching an r-handle. The boundary 
is obtained from 
by surgery. Morse theory was used by Thom and Milnor to prove that every manifold (with or without boundary) is a handlebody, meaning that it has an expression as a union of handles. The expression is non-unique: the manipulation of handlebody decompositions is an essential ingredient of the proof of the Smale h-cobordism theorem, and its generalization to the s-cobordism theorem.
[edit] 3-dimensional handlebodies
A handlebody can be defined as an orientable 3-manifold-with-boundary containing 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.)
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
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.
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.
