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

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

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

Compression bodies often arise when manipulating Heegaard splittings.