Atoroidal

• 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 to avoid being one of three kinds of fiber bundle. He makes the same restriction on geometrically atoroidal manifolds 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 manifolds.