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.

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.

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.

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 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.

Some famous examples of 3-manifolds

*3-sphere
*SO(3) (or 3-dimensional real projective space)
*3-torus
*hyperbolic 3-space
*Poincaré dodecahedral space
*Seifert-Weber space

Some important classes of 3-manifolds

*knot and link complements
*Haken manifold
*lens space
*Seifert fiber spaces
*graph manifold
*hyperbolic 3-manifold
*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
*Contact structure

Foundational results

*Prime decomposition theorem
*Moise's theorem - Every 3-manifold has a triangulation, unique up to common subdivision
**As corollary, every compact 3-manifold has a Heegaard splitting.
*JSJ decomposition, also known as the toral decomposition
*Kneser-Haken finiteness
*Loop and sphere theorems
*Lickorish-Wallace theorem
*Waldhausen theorems on topological rigidity
*Thurston's hyperbolic Dehn surgery theorem
*Thurston's geometrization theorem for Haken manifolds
*Scott core theorem
*Cyclic surgery theorem

Some famous conjectures

*Poincaré conjecture
*Thurston's geometrization conjecture
*Virtually fibered conjecture
*Virtually Haken conjecture
*Cabling conjecture
*Surface subgroup conjecture
*Simple loop conjecture
*Waldhausen conjecture on Heegaard splittings

References

*Hempel, 3-manifolds, American Mathematical Society, ISBN 0821836951
*Jaco, Lectures on three-manifold topology, American Mathematical Society, ISBN 082181693
*Rolfsen, Knots and Links, American Mathematical Society, ISBN 082183436
*Thurston, Three-dimensional geometry and topology, Princeton University Press, ISBN 0691083045
*Adams, The Knot Book, American Mathematical Society, ISBN 0805073809
*Hatcher, Notes on basic 3-manifold topology, available online
*R. H. Bing, The Geometric Topology of 3-Manifolds, (1983) American Mathematical Society Colloquium Publications Volume 40, Providence RI, ISBN 0-8218-1040-5.