Handlebody

In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.

Handlebodies play a similar role in the study of manifolds as simplicial complexes and CW complexes play in homotopy theory, allowing one to analyze a space in terms of individual pieces and their interactions.

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.

Any handlebody decomposition of a manifold defines a CW complex decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell.

However, a handlebody decomposition gives more information than just the homotopy type of the manifold.

For instance, a handlebody decomposition completely describes the manifold up to homeomorphism.

This is false in higher dimensions; any exotic sphere is the union of a 0-handle and an n-handle.

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.

(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 importance of handlebodies in 3-manifold theory comes from their connection with Heegaard splittings.

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.