Handlebody

Summary

In 3-manifold topology, a handlebody of genus $g$ is the compact orientable $3$-manifold obtained from a $3$-ball by attaching $g$ one-handles. It is a solid version of the closed orientable surface of genus $g$, which is its boundary. It is a ball for $g = 0$, a solid torus for $g = 1$, and a solid double torus for $g = 2$. Every closed orientable $3$-manifold is obtained by gluing two handlebodies of the same genus along their boundaries, which is a Heegaard splitting. More generally, the word handlebody is also used for any manifold with a given handle decomposition.