• Source: 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


{\displaystyle S}

be a compact, closed surface (not necessarily connected). Attach 1-handles to



S
×
[
0
,
1
]


{\displaystyle S\times [0,1]}

along



S
×
{
1
}


{\displaystyle S\times \{1\}}

.
Let



C


{\displaystyle C}

be a compression body. The negative boundary of C, denoted




∂

−


C


{\displaystyle \partial _{-}C}

, is



S
×
{
0
}


{\displaystyle S\times \{0\}}

. (If



C


{\displaystyle C}

is a handlebody then




∂

−


C
=
∅


{\displaystyle \partial _{-}C=\emptyset }

.) The positive boundary of C, denoted




∂

+


C


{\displaystyle \partial _{+}C}

, is



∂
C


{\displaystyle \partial C}

minus the negative boundary.
There is a dual construction of compression bodies starting with a surface



S


{\displaystyle S}

and attaching 2-handles to



S
×
{
0
}


{\displaystyle S\times \{0\}}

. In this case




∂

+


C


{\displaystyle \partial _{+}C}

is



S
×
{
1
}


{\displaystyle S\times \{1\}}

, and




∂

−


C


{\displaystyle \partial _{-}C}

is



∂
C


{\displaystyle \partial C}

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. (eds.). Handbook of Geometric Topology. North-Holland. pp. 93–164. MR 1886669.

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