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

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

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

Compression bodies often arise when manipulating Heegaard splittings.

References

• Bonahon, Francis (2002). "Geometric structures on 3-manifolds". in Daverman, Robert J.; Sher, Richard B.. Handbook of Geometric Topology. North-Holland. pp. 93–164. 

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