# Atoroidal

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In [mathematics](/source/mathematics), an '''atoroidal''' [3-manifold](/source/3-manifold) is one that does not contain an essential [torus](/source/torus).
There are two major variations in this terminology: an essential torus may be defined geometrically, as an [embedded](/source/embedding), non-[boundary parallel](/source/boundary_parallel),  [incompressible](/source/Incompressible_surface) [torus](/source/torus), or it may be defined algebraically, as a [subgroup](/source/subgroup) <math>\Z\times\Z</math> of its [fundamental group](/source/fundamental_group) that is not [conjugate](/source/Conjugacy_class) to a [peripheral subgroup](/source/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:
*{{harvs|txt|last=Apanasov|first=Boris|year=2000}} 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](/source/Prime_decomposition_(3-manifold)) [boundary-incompressible](/source/Boundary-incompressible_surface) 3-manifolds this gives the algebraic definition.<ref>{{citation|title=Conformal Geometry of Discrete Groups and Manifolds|volume=32|series=De Gruyter Expositions in Mathematics|first=Boris N.|last=Apanasov|publisher=[Walter de Gruyter](/source/Walter_de_Gruyter)|year=2000|isbn=9783110808056|page=294|url=https://books.google.com/books?id=Y-aIVhfbIugC&pg=PA294}}.</ref>
*{{harvs|txt|last=Otal|first=Jean-Pierre|year=2001}} uses the algebraic definition without additional restrictions.<ref>{{citation|title=The hyperbolization theorem for fibered 3-manifolds|volume=7|series=Contemporary Mathematics|first=Jean-Pierre|last=Otal|publisher=[American Mathematical Society](/source/American_Mathematical_Society)|year=2001|isbn=9780821821534|page=ix|url=https://books.google.com/books?id=pVObtYVehxIC&pg=PR9}}.</ref>
*{{harvs|txt|last=Chow|first=Bennett|year=2007}} uses the geometric definition, restricted to irreducible manifolds.<ref>{{citation|title=The Ricci Flow: Geometric aspects|series=Mathematical surveys and monographs|first=Bennett|last=Chow|publisher=[American Mathematical Society](/source/American_Mathematical_Society)| year=2007|isbn=9780821839461|page=436|url=https://books.google.com/books?id=T3gqWWbCd60C&pg=PA436}}.</ref>
*{{harvs|txt|first=Michael|last=Kapovich|authorlink=Michael Kapovich|year=2009}} requires the algebraic variant of atoroidal manifolds (which he calls simply atoroidal) to avoid being one of three kinds of [fiber bundle](/source/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 bottle](/source/Klein_bottle)s. With these definitions, the two kinds of atoroidality are equivalent except on certain [Seifert manifold](/source/Seifert_manifold)s.<ref>{{citation|title=Hyperbolic Manifolds and Discrete Groups|volume=183|series=Progress in Mathematics| first=Michael|last=Kapovich|authorlink=Michael Kapovich|publisher=Springer|year=2009|isbn=9780817649135|page=6|url=https://books.google.com/books?id=JRJ8VmfP-hcC&pg=PA6}}.</ref>

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

==References==
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Category:3-manifolds

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Adapted from the Wikipedia article [Atoroidal](https://en.wikipedia.org/wiki/Atoroidal) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Atoroidal?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.