Compression body
Template:One source 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>S</math> be a compact, closed surface (not necessarily connected). Attach 1-handles to <math>S \times [0,1]</math> along <math>S \times \{1\}</math>.
Let <math>C</math> be a compression body. The negative boundary of C, denoted <math>\partial_{-}C</math>, is <math>S \times \{0\}</math>. (If <math>C</math> is a handlebody then <math>\partial_- C = \emptyset</math>.) The positive boundary of C, denoted <math>\partial_{+}C</math>, is <math>\partial C</math> minus the negative boundary.
There is a dual construction of compression bodies starting with a surface <math>S</math> and attaching 2-handles to <math>S \times \{0\}</math>. In this case <math>\partial_{+}C</math> is <math>S \times \{1\}</math>, and <math>\partial_{-}C</math> is <math>\partial C</math> minus the positive boundary.
Compression bodies often arise when manipulating Heegaard splittings.