In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, there are three definitions for atoroidal as applied to 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

s:

• A 3-manifold is (geometrically) atoroidal if it does not contain an embedded
  Embedding
  In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
  
  , non-boundary parallel
  Boundary parallel
  In mathematics, a closed n-manifold N embedded in an -manifold M is boundary parallel if there is an isotopy of N onto a boundary component of M.-An example:...
  
  , incompressible
  Incompressible surface
  In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
  
   torus
  Torus
  In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
  
  .
• A 3-manifold is (geometrically) atoroidal if both of the following hold:
  • It does not contain an embedded, non-boundary parallel, incompressible torus.
  • It is acylindrical (also called anannular), meaning that it does not contain a properly embedded, non-boundary parallel, incompressible annulus
    Annulus (mathematics)
    In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
    
    .
• A 3-manifold is (algebraically) atoroidal if any subgroup
  Subgroup
  In group theory, 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 of G if the restriction of * to H x H is a group operation on H...
  
    of its fundamental group
  Fundamental group
  In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
  
   is conjugate to a peripheral subgroup, i.e. the image of the map on fundamental group induced by an inclusion of a boundary component.

Any algebraically atoroidal 3-manifold is geometrically atoroidal; but the converse

Converse (logic)

In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P. For the categorical proposition All S is P, the converse is All P is S. In neither case does the converse necessarily follow from...

is false. However, the mathematical literature often fails to distinguish between them, so one must ascertain any given author's intent.

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