Information About3-manifold |
| CATEGORIES ABOUT 3-MANIFOLD | |
| geometric topology | |
| 3-manifolds | |
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Phenomena in three dimensions can be strikingly different than that for other dimenions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. Perhaps surprisingly, this special role has led to the discovery of close connections to a diversity of other fields, such as Knot Theory , Geometric Group Theory , Hyperbolic Geometry , Number Theory , Teichmüller Theory , Topological Quantum Field Theory , Gauge Theory , Floer Homology , and Partial Differential Equations . 3-manifold theory is considered a part of Low-dimensional Topology or Geometric Topology . 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. SOME FAMOUS EXAMPLES OF 3-MANIFOLDS
Hyperbolic link complements The following examples are particularly well-known and studied. SOME IMPORTANT CLASSES OF 3-MANIFOLDS
The classes are not necessarily mutually exclusive! SOME IMPORTANT STRUCTURES ON 3-MANIFOLDS FOUNDATIONAL RESULTS Some results are named as conjectures as a result of historical artifacts.
SOME FAMOUS CONJECTURES Some of these are thought to be solved, as of March 2006. Please see specific articles for more information.
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