In mathematics, an atoroidal3-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, incompressibletorus, or it may be defined algebraically, as a subgroup 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 (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 irreducibleboundary-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]
Michael Kapovich (2009) requires the algebraic variant of atoroidal manifolds (which he calls simply atoroidal) to avoid being one of three kinds of 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 bottles. With these definitions, the two kinds of atoroidality are equivalent except on certain Seifert manifolds.[4]
A 3-manifold that is not atoroidal is called toroidal.
References
^Apanasov, Boris N. (2000), Conformal Geometry of Discrete Groups and Manifolds, De Gruyter Expositions in Mathematics, vol. 32, Walter de Gruyter, p. 294, ISBN9783110808056.
atoroidal, mathematics, atoroidal, manifold, that, does, contain, essential, torus, there, major, variations, this, terminology, essential, torus, defined, geometrically, embedded, boundary, parallel, incompressible, torus, defined, algebraically, subgroup, di. 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 Z Z 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 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 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 Michael Kapovich 2009 requires the algebraic variant of atoroidal manifolds which he calls simply atoroidal to avoid being one of three kinds of 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 bottles With these definitions the two kinds of atoroidality are equivalent except on certain Seifert manifolds 4 A 3 manifold that is not atoroidal is called toroidal References Edit Apanasov Boris N 2000 Conformal Geometry of Discrete Groups and Manifolds De Gruyter Expositions in Mathematics vol 32 Walter de Gruyter p 294 ISBN 9783110808056 Otal Jean Pierre 2001 The hyperbolization theorem for fibered 3 manifolds Contemporary Mathematics vol 7 American Mathematical Society p ix ISBN 9780821821534 Chow Bennett 2007 The Ricci Flow Geometric aspects Mathematical surveys and monographs American Mathematical Society p 436 ISBN 9780821839461 Kapovich Michael 2009 Hyperbolic Manifolds and Discrete Groups Progress in Mathematics vol 183 Springer p 6 ISBN 9780817649135 This geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Atoroidal amp oldid 897199067, wikipedia, wiki, book, books, library,
