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3-manifold

A key idea in the theory is to study a 3-manifold by considering special [surface] s embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an [incompressible surface] and the theory of [Haken manifold] s, or one can choose the complementary pieces to be as nice as possible, leading to structures such as [Heegaard splitting] s, which are useful even in the non-Haken case.

[Thurston's] contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is [hyperbolic geometry] . Using a geometry in addition to special surfaces is often fruitful.

The [fundamental group] s of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between [group theory] and topological methods.

Important examples of 3-manifolds

- [Euclidean 3-space]
- [3-sphere]
- [SO(3)] (or 3-dimensional [real projective space] )
- [3-torus]
- [Hyperbolic 3-space]
- [Poincaré dodecahedral space]
- [Seifert-Weber space]
- [Gieseking manifold]

Hyperbolic link complements
The following examples are particularly well-known and studied.

- [Figure eight knot]
- [Whitehead link]
- [Borromean rings]

Some important classes of 3-manifolds

- [Graph manifold]
- [Haken manifold]
- [Homology sphere] s
- [Hyperbolic 3-manifold]
- [I-bundle] s
- [Knot and link complements]
- [Lens space]
- [Seifert fiber spaces] , [Circle bundle] s
- [Spherical 3-manifold]
- [Surface bundles over the circle]
- [Torus bundle]

The classes are not necessarily mutually exclusive!

Some important structures on 3-manifolds

- [Heegaard splitting]
- [Essential lamination]
- [Taut foliation]
- [Haken hierarchy]

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