Handlebody

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.

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.

If ${\displaystyle (W,\partial W)}$ is an ${\displaystyle n}$-dimensional manifold with boundary, and

is an embedding, the ${\displaystyle n}$-dimensional manifold with boundary

...
Wikipedia