In mathematics, an atoroidal

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

is one that does not contain an essential

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non- boundary parallel,

incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...

, or it may be defined algebraically, as a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

\Z\times\Z

of its fundamental group that is not conjugate to a peripheral subgroup (i.e., the image of the map on fundamental group induced by an inclusion of a boundary component). The terminology is not standardized, and different authors require atoroidal 3-manifolds to satisfy certain additional restrictions. For instance: * gives a definition of atoroidality that combines both geometric and algebraic aspects, in terms of maps from a torus to the manifold and the induced maps on the fundamental group. He then notes that for irreducible boundary-incompressible 3-manifolds this gives the algebraic definition. * uses the algebraic definition without additional restrictions. * uses the geometric definition, restricted to irreducible manifolds. * requires the algebraic variant of atoroidal manifolds (which he calls simply atoroidal) to avoid being one of three kinds of fiber bundle. He makes the same restriction on geometrically atoroidal manifolds (which he calls topologically atoroidal) and in addition requires them to avoid incompressible boundary-parallel embedded Klein bottles. With these definitions, the two kinds of atoroidality are equivalent except on certain

Seifert manifold A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...

s.. A 3-manifold that is not atoroidal is called toroidal.

References

{{geometry-stub 3-manifolds