Incompressible surface

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.

If D were to bound a disk inside S (which is always the case if S is incompressible, for example), then compressing S along D would result in a disjoint union of a sphere and a surface homeomorphic to S. The resulting surface with the sphere deleted might or might not be isotopic to S, and it will be if S is incompressible and M is irreducible.

Then S is π1-injective (or algebraically incompressible) if the induced map on fundamental groups is injective.

In general, every π1-injective surface is incompressible, but the reverse implication is not always true.

For instance, the Lens space L(4,1) contains an incompressible Klein bottle that is not π1-injective.

However, if S is two-sided, the loop theorem implies Kneser's lemma, that if S is incompressible, then it is π1-injective.

David Gabai proved in particular that a genus-minimizing Seifert surface is a leaf of some taut, transversely oriented foliation of the knot complement, which can be certified with a taut sutured manifold hierarchy.

The two maps from π1(S) into π1(S3 − N(S)) given by pushing loops off the surface to the positive or negative side of N(S) are both injections.