- 3-manifold
In
mathematics , a 3-manifold is a 3-dimensionalmanifold . 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.Phenomena in three dimensions can be strikingly different from that for other dimensions, 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 , andpartial differential equations . 3-manifold theory is considered a part oflow-dimensional topology orgeometric 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 anincompressible surface and the theory ofHaken manifold s, or one can choose the complementary pieces to be as nice as possible, leading to structures such asHeegaard 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 betweengroup theory and topological methods.Important examples of 3-manifolds
*
Euclidean 3-space
*3-sphere
*SO(3) (or 3-dimensionalreal 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
*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
*Trigenus Foundational results
Some results are named as conjectures as a result of historical artifacts.
We begin with the purely topological:
*
Moise's theorem - Every 3-manifold has a triangulation, unique up to common subdivision
** As corollary, every compact 3-manifold has aHeegaard splitting .
* Prime decomposition theorem
* Kneser-Haken finiteness
* Loop and sphere theorems
* Annulus andTorus theorem
*JSJ decomposition , also known as the toral decomposition
*Scott core theorem
*Lickorish-Wallace theorem
* Waldhausen's theorems on topological rigidity
*Waldhausen conjecture on Heegaard splittingsTheorems where geometry plays an important role in the proof:
*
Smith conjecture
*Cyclic surgery theorem Results explicitly linking geometry and topology:
* Thurston's
hyperbolic Dehn surgery theorem
* Jorgensen-Thurston theorem that the set of finite volumes of hyperbolic 3-manifolds has order type omega^omega.
* Thurston'sgeometrization theorem for Haken manifolds
*Tameness conjecture , also called the Marden conjecture or tame ends conjecture
*Ending lamination conjecture Important conjectures
Some of these are thought to be solved, as of March 2007. Please see specific articles for more information.
*
Poincaré conjecture — see alsoSolution of the Poincaré conjecture
*Thurston's geometrization conjecture
*Virtually fibered conjecture
*Virtually Haken conjecture
*Cabling conjecture
*Surface subgroup conjecture
*Simple loop conjecture
* The smallest hyperbolic 3-manifold is theWeeks manifold .
*Lubotzy-Sarnak conjecture onproperty tau References
* Hempel, "3-manifolds", American Mathematical Society, ISBN 0-8218-3695-1
* Jaco, "Lectures on three-manifold topology", American Mathematical Society, ISBN 0-8218-1693-4
* Rolfsen, "Knots and Links", American Mathematical Society, ISBN 0-914098-16-0
* Thurston, "Three-dimensional geometry and topology", Princeton University Press, ISBN 0-691-08304-5
* Adams, "The Knot Book", American Mathematical Society, ISBN 0-8050-7380-9
* Hatcher, "Notes on basic 3-manifold topology", [http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html 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.
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