Incompressible surface

In mathematics, incompressible surfaces are an important tool of 3-dimensional topology. By cutting along incompressible surfaces 3 - dimensional manifolds can be decomposed into simpler pieces.

Definition

Let be a 3 -dimensional manifold with (possibly empty ) boundary and a 2 -dimensional manifold, ie one actually embedded surface.

Incompressible surface

A compression disk for is an embedded disc

So that in the not homotopic to a constant map is.

The surface is called incompressible if

• And there is no compression on disc, or
• And is not homotopic to a constant map in.

Boundary incompressible surface

An edge compression disc for is with an embedded triples, so not ( rel.) isotope to a Einbettunng with image is the image and each in circular discs cut.

The surface is non-compressible, if there is no margin for compression disc.

Fundamental group

When a non-compressible surface is, then the current induced by the inclusion of the fundamental group homomorphism

Injective. For two -sided faces the converse also holds: a coherent two-sided surface is incompressible if and only if it is injective.

Existence

If a compact irreducible 3-manifold, then there is every homology class

A ( directional, possibly discontinuous ) non-compressible and non-compressible surface, such that

Herein, the inclusion and the fundamental class of the.

Set of hooks

The set of hooks states that cutting a 3-manifold along an incompressible, edge - incompressible surface reduces the complexity of the hook 3-manifold. This is often used in the three -dimensional topology to perform proofs by induction on the hook complexity.

Minimal surfaces

By a theorem of Freedman, Hass and Scott every incompressible surface ( in a compact 3-manifold ) isotopic to a minimal surface from index 0