Compression body

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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.

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Compression body

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