Editing Incompressible surface (section)

== Formal definition ==

[[File:Compressing disk in an incompressible surface.svg|right|400px|thumb|For an incompressible surface {{math|''S''}}, every compressing disk {{math|''D''}} bounds a disk {{math|''D&prime;''}} in {{math|''S''}}.  Together, {{math|''D''}} and {{math|''D&prime;''}} form a 2-sphere.  This sphere need not bound a ball unless {{math|''M''}} is [[Prime manifold#Irreducible manifold|irreducible]].]]

Let {{math|''S''}} be a [[compact surface]] properly embedded in a [[smooth manifold|smooth]] or [[piecewise linear manifold|PL]] 3-manifold {{math|''M''}}.  A '''compressing disk''' {{math|''D''}} is a [[disk (mathematics)|disk]] embedded in {{math|''M''}} such that

:<math>D \cap S = \partial D</math>

and the intersection is [[Transversality (mathematics)|transverse]].  If the curve {{math|&part;''D''}} does not bound a disk inside of {{math|''S''}}, then {{math|''D''}} is called a '''nontrivial''' compressing disk.  If {{math|''S''}} has a nontrivial compressing disk, then we call {{math|''S''}} a '''compressible''' surface in {{math|''M''}}.

If {{math|''S''}} is neither the [[2-sphere]] nor a compressible surface, then we call the surface ('''geometrically''') '''incompressible'''.

Note that 2-spheres are excluded since they have no nontrivial compressing disks by the [[Schoenflies problem|Jordan-Schoenflies theorem]], and 3-manifolds have abundant embedded 2-spheres.  Sometimes one alters the definition so that an '''incompressible sphere''' is a 2-sphere embedded in a 3-manifold that does not bound an embedded [[Ball (mathematics)|3-ball]].  Such spheres arise exactly when a 3-manifold is not [[Prime manifold#Irreducible manifold|irreducible]].  Since this notion of incompressibility for a sphere is quite different from the above definition for surfaces, often an incompressible sphere is instead referred to as an '''essential sphere''' or a '''reducing sphere'''.