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.

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, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology.

Modeling the universe as a 3-manifold

An important scientific application of 3-manifolds is in physical cosmology, as models for the Shape of the Universe – the surface of the earth is locally approximately flat – it is roughly a 2-manifold, and globally the surface of the earth is a sphere. The universe, likewise, looks locally approximately like 3-dimensional Euclidean space, so the universe may be modeled as a 3-manifold, and one may ask which 3-manifold it is. In physical cosmology, spacetime is typically assumed to have a decomposition into a 3-dimensional spacial manifold and one dimension of time.

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.

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 spheres
• Hyperbolic 3-manifold
• I-bundles
• Knot and link complements
• Lens space
• Seifert fiber spaces, Circle bundles
• 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
• 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 a Heegaard splitting.
• Prime decomposition theorem
• Kneser-Haken finiteness
• Loop and sphere theorems
• Annulus and Torus 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 splittings

Theorems 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 ω^ω.
• Thurston's geometrization 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 also Solution 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 the Weeks manifold.
• Lubotzy-Sarnak conjecture on property tau

References

• Hempel, John (2004), 3-manifolds, Providence, RI: American Mathematical Society, ISBN 0821836951 
• Jaco, William H. (1980), Lectures on three-manifold topology, Providence, RI: American Mathematical Society, ISBN 0821816934 
• Rolfsen, Dale (1976), Knots and Links, Providence, RI: American Mathematical Society, ISBN 0914098160 
• Thurston, William P. (1997), Three-dimensional geometry and topology, Princeton, NJ: Princeton University Press, ISBN 0691083045 
• Adams, Colin Conrad (2002), The Knot Book, New York: W. H. Freeman, ISBN 0805073809 
• Bing, R. H. (1983), The Geometric Topology of 3-Manifolds, Colloquium Publications, 40, Providence, RI: American Mathematical Society, ISBN 0821810405 

External links

• Hatcher, Allen, Notes on basic 3-manifold topology, Cornell University, http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html