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.

References

F.Bonahon, Geometric structures on 3-manifolds, Handbook of Geometric Topology, Daverman and Sher eds. North-Holland (2002).

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