mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an atoroidal

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

is one that does not contain an essential

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

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

boundary parallel In mathematics, a boundary parallel, ∂-parallel, or peripheral closed ''n''-manifold ''N'' embedded in an (''n'' + 1)-manifold ''M'' is one for which there is an isotopy of ''N'' onto a boundary component of ''M''.Definition 3.4.7 ...

incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...

, or it may be defined algebraically, as a

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

\Z\times\Z

of its

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

that is not conjugate to a

peripheral subgroup In algebraic topology, a peripheral subgroup for a topological pair, space-subspace pair ''X'' ⊃ ''Y'' is a certain subgroup of the fundamental group of the complementary space, π1(''X'' − ''Y''). Its conjugacy class is an ...

(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 In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole. Emergence plays a central role ...

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 In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...

. 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 bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...

s. 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 al ...

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

References

3-manifolds {{geometry-stub