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In mathematics, there are three definitions for atoroidal as applied to 3-manifolds:

- A 3-manifold is (geometrically) atoroidal if it does not contain an embedded, non-boundary parallel, incompressible torus.
- 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.
- A 3-manifold is (algebraically) atoroidal if any subgroup $mathbb\; Ztimesmathbb\; Z$ of its fundamental group 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 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.

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Last updated on Thursday February 22, 2007 at 14:00:34 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday February 22, 2007 at 14:00:34 PST (GMT -0800)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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