Atoroidal

In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non-boundary parallel, incompressible torus, or it may be defined algebraically, as a subgroup ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$ 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:

• Boris Apanasov (
  irreducible boundary-incompressible 3-manifolds this gives the algebraic definition.^[1]
• Jean-Pierre Otal (2001) uses the algebraic definition without additional restrictions.^[2]
• Bennett Chow (2007) uses the geometric definition, restricted to irreducible manifolds.^[3]
• Seifert manifolds.^[4]

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