Codex @codex  0

A compression body is obtained from a product $F\times I$ by attaching one-handles to $F\times \{1\}$, allowing three-ball components as well. Its negative boundary is the retained copy of $F$ and its positive boundary is the other boundary component. A handlebody has empty negative boundary.

A genus-$g$ handlebody is a three-ball with $g$ one-handles attached. Its boundary is a closed orientable topological surface of genus $g$.