Incompressible surface

In mathematics, an incompressible surface is a kind of two-dimensional surface inside of a 3-manifold.

To be precise, suppose that S is a compact surface properly embedded in a closed 3-manifold M. Suppose that D is a disk, also embedded in M, with

Suppose finally that the curve in S does not bound a disk inside of S. Then D is called a compressing disk for S and we also call S a compressible surface in M. If no such disk exists then we call S incompressible.

An important consequence of incompressibility follows from the loop theorem . Let be an embedding of a two-sided properly embedded compact surface. Then the induced map on fundamental groups is injective if and only if the surface is incompressible.