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
×
[
0
,
1
]
along
S
×
{
1
}
.
Let
C
be a compression body. The negative boundary of C, denoted
∂
−
C
, is
S
×
{
0
}
. (If
C
is a handlebody then
∂
−
C
=
∅
.) The positive boundary of C, denoted
∂
+
C
, is
∂
C
minus the negative boundary.
There is a dual construction of compression bodies starting with a surface
S
and attaching 2-handles to
S
×
{
0
}
. In this case
∂
+
C
is
S
×
{
1
}
, and
∂
−
C
is
∂
C
minus the positive boundary.
Compression bodies often arise when manipulating Heegaard splittings.
