In the theory of

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

s, a compression body is a kind of generalized

handlebody In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Handles ...

. A compression body is either a

or the result of the following construction: : Let

S

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

S \times,1 /math> along S \times \ .

Let C be a compression body. The negative boundary of C, denoted \partial_C, is S \times \ .  (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 \ . In this case \partial_C is S \times \, and \partial_C is \partial C minus the positive boundary.

Compression bodies often arise when manipulating

Heegaard splitting In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Definitions Let ''V'' and ''W'' be handlebodies of genus ''g'', and ...

References

* {{cite book, editor1-link=Robert Daverman, first=Francis, last=Bonahon, contribution= Geometric structures on 3-manifolds, pages=93–164, title=Handbook of Geometric Topology, editor1-last= Daverman , editor1-first=Robert J., editor2-last= Sher , editor2-first=Richard B., publisher= North-Holland , year=2002, MR=1886669 3-manifolds de:Henkelkörper#Kompressionskörper