3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.

Related Topics:
Mathematics - Manifold

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The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds. There are close connections to other fields, such as 4-manifolds, surfaces, knot theory, topological quantum field theory, and gauge theory. 3-manifold theory is a part of low-dimensional topology or geometric topology.

Related Topics:
4-manifold - Surface - Knot theory - Topological quantum field theory - Gauge theory - Low-dimensional topology - Geometric topology

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A key idea in the theory is to study a 3-manifold by considering special surfaces 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 manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful even in the non-Haken case.

Related Topics:
Surface - Incompressible surface - Haken manifold - Heegaard splitting

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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.

Related Topics:
Thurston's - Hyperbolic geometry

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The fundamental groups 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.

Related Topics:
Fundamental group - Group theory

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