Incompressible area

In mathematics , incompressible surfaces are an important aid in 3-dimensional topology. By cutting open along incompressible surfaces, 3-dimensional manifolds can be broken down into simpler pieces.

definition

Let be a 3-dimensional manifold with an (possibly empty) margin and a 2-dimensional submanifold , i.e. H. an actually embedded surface. ${\displaystyle (M,\partial M)}$ ${\displaystyle (F,\partial F)\subset (M,\partial M)}$

Incompressible area

A compression disk for is an embedded circular disk ${\displaystyle F}$

${\displaystyle (D^{2},\partial D^{2})\subset (M,F)}$,

so that in not homotop is a constant figure. ${\displaystyle F\cap D^{2}=\partial D^{2}}$ ${\displaystyle F}$

The area is called incompressible if ${\displaystyle F}$

• ${\displaystyle F\not =S^{2},D^{2},\mathbb {R} P^{2}}$and there is no compression disk for , or ${\displaystyle F}$
• ${\displaystyle F=D^{2}}$and is in non- homotop to a constant figure. ${\displaystyle \partial F}$ ${\displaystyle \partial M}$

Edge-incompressible area

An edge-compression disc for an embedded triples with , so that no (rel. ) Isotope to an embedment of image is, the image and each cut into circular discs. ${\displaystyle F}$ ${\displaystyle (D^{2},A,B)\subset (M,\partial M,F)}$ ${\displaystyle A\cup B=\partial D^{2},D^{2}\cap F=B}$ ${\displaystyle D^{2}}$ ${\displaystyle \partial D^{2}}$ ${\displaystyle \partial M\cup F}$ ${\displaystyle \partial M}$ ${\displaystyle F}$

The area is called - incompressible if there is no edge compression disk for . ${\displaystyle F}$ ${\displaystyle \partial }$ ${\displaystyle F}$

In the case of manifolds with a non-empty edge, the term incompressible surfaces is often used when surfaces are meant that are incompressible and edge-incompressible in the sense of the above definitions .

Fundamental group

If an incompressible surface is in, then the inclusion induced homomorphism is the fundamental groups ${\displaystyle F}$ ${\displaystyle M}$ ${\displaystyle i:F\rightarrow M}$

${\displaystyle i_{*}:\pi _{1}F\rightarrow \pi _{1}M}$

injective . The converse also applies to two-sided surfaces : a connected two-sided surface is incompressible if and only if it is -injective. ${\displaystyle \pi _{1}}$

existence

If is a compact irreducible 3-manifold, then for every class there is homology ${\displaystyle M}$

${\displaystyle z\in H_{2}(M,\partial M;\mathbb {Z} )}$

an (orientable, possibly incoherent) incompressible and incompressible surface , so that ${\displaystyle \partial }$ ${\displaystyle F\subset M}$

${\displaystyle i_{*}\left[F,\partial F\right]=z}$.

Here denotes the inclusion and the fundamental class of . ${\displaystyle i:(F,\partial F)\rightarrow (M,\partial M)}$ ${\displaystyle \left[F,\partial F\right]\in H_{2}(F,\partial F;\mathbb {Z} )}$ ${\displaystyle F}$

Set of hooks

The theorem of hooks states that cutting open a 3-manifold along an incompressible, edge-incompressible surface reduces the hook-complexity of the 3-manifold. This is often used in 3-dimensional topology to induce proofs on the hook complexity.

Minimal areas

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

See also

• Hook manifold

literature

• William Jaco : Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43rd American Mathematical Society, Providence, RI, 1980. ISBN 0-8218-1693-4