Incompressible surface

In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a 'nontrivial' surface that cannot be simplified by pinching off tubes. They are useful for decomposition of Haken manifolds, normal surface theory, and studying fundamental groups of 3-manifolds. In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a 'nontrivial' surface that cannot be simplified by pinching off tubes. They are useful for decomposition of Haken manifolds, normal surface theory, and studying fundamental groups of 3-manifolds. Let S be a compact surface properly embedded in a smooth or PL 3-manifold M. A compressing disk D is a disk embedded in M such that and the intersection is transverse. If the curve ∂D does not bound a disk inside of S, then D is called a nontrivial compressing disk. If S has a nontrivial compressing disk, then we call S a compressible surface in M. If 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 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 3-ball. Such spheres arise exactly when a 3-manifold is not 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. Given a compressible surface S with a compressing disk D that we may assume lies in the interior of M and intersects S transversely, one may perform embedded 1-surgery on S to get a surface that is obtained by compressing S along D. There is a tubular neighborhood of D whose closure is an embedding of D × with D × 0 being identified with D and with