Atoroidal manifold

In three-dimensional topology , atoroidality describes a relationship between the edge of a manifold and the manifold itself.

An irreducible manifold is called geometrically atoroidal if every 2- torus embedded in incompressibly can be shifted by an isotopy to an edge component of . This means that it does not contain any embedded tori other than those that obviously must exist. ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$

An irreducible manifold is called homotopic atoroidal if every mapping that injectively maps the fundamental group of the torus into the fundamental group of is homotopic to a mapping into the boundary. This corresponds to the property of the fundamental group of that every subgroup of the form is conjugate to the fundamental group of a torus boundary component. ${\displaystyle T^{2}\to M}$ ${\displaystyle T^{2}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$

One can show that "geometrically atoroidal" follows from "homotopic atoroidal". However, the reverse is not true. The torus theorem says that a geometrically atoroidal 3-manifold is either homotopically atoroidal or a Seifertian fiber space .

The Hyperbolisierungvermutung Thurston says that any irreducible homotopisch atoroidale manifold carries a hyperbolic structure with infinite fundamental group.

Individual evidence

1. ↑ P. Scott: A new proof of the annulus and torus theorems. Amer. J. Math. 102 (1980) no. 2, 241-277.