Compression body

In the theory of 3-manifolds, a compression body is a kind of generalized handlebody.

A compression body is either a handlebody or the result of the following construction:

Let ${\displaystyle S}$ be a compact, closed surface (not necessarily connected). Attach 1-handles to ${\displaystyle S\times [0,1]}$ along ${\displaystyle S\times \{1\}}$.

Let ${\displaystyle C}$ be a compression body. The negative boundary of C, denoted ${\displaystyle \partial _{-}C}$, is ${\displaystyle S\times \{0\}}$. (If ${\displaystyle C}$ is a handlebody then ${\displaystyle \partial _{-}C=\emptyset }$.) The positive boundary of C, denoted ${\displaystyle \partial _{+}C}$, is ${\displaystyle \partial C}$ minus the negative boundary.

There is a dual construction of compression bodies starting with a surface ${\displaystyle S}$ and attaching 2-handles to ${\displaystyle S\times \{0\}}$. In this case ${\displaystyle \partial _{+}C}$ is ${\displaystyle S\times \{1\}}$, and ${\displaystyle \partial _{-}C}$ is ${\displaystyle \partial C}$ minus the positive boundary.

Compression bodies often arise when manipulating Heegaard splittings.