Incompressible surface

Summary

An incompressible surface is a surface inside a 3-manifold that cannot be simplified by cutting it along a disk in the manifold. For two-sided surfaces other than spheres and disks, this means that the inclusion of the surface induces an injective map on fundamental groups. Incompressible surfaces are the main tool to cut $3$-manifolds into simpler pieces, as in the theory of Haken manifolds and the JSJ decomposition.