3-manifold
Sign in to saveAlso known as three-manifold
right|thumb|250px| An image from inside a Three-torus|3-torus. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.
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- 3-manifolds
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~30 min read
Article
50 sectionsContents
- Principles
- Definition
- Mathematical theory of 3-manifolds
- Invariants describing 3-manifolds
- Connected sums
- Second homotopy groups
- Important examples of 3-manifolds
- Euclidean 3-space
- 3-sphere
- Real projective 3-space
- 3-torus
- Hyperbolic 3-space
- Poincaré dodecahedral space
- Seifert–Weber space
- Gieseking manifold
- Some important classes of 3-manifolds
- Hyperbolic link complements
- Some important structures on 3-manifolds
- Contact geometry
- Haken manifold
- Essential lamination
- Heegaard splitting
- Taut foliation
- Foundational results
- Moise's theorem
- Prime decomposition theorem
- Kneser–Haken finiteness
- Loop and Sphere theorems
- Annulus and Torus theorems
- JSJ decomposition
- Scott core theorem
- Lickorish–Wallace theorem
- Waldhausen's theorems on topological rigidity
- Waldhausen conjecture on Heegaard splittings
- Smith conjecture
- Cyclic surgery theorem
- Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem
- Thurston's hyperbolization theorem for Haken manifolds
- Tameness conjecture, also called the Marden conjecture or tame ends conjecture
- Ending lamination conjecture
- Poincaré conjecture
- Thurston's geometrization conjecture
- Virtually fibered conjecture and Virtually Haken conjecture
- Simple loop conjecture
- Surface subgroup conjecture
- Important conjectures
- Cabling conjecture
- References
- Further reading
- External links
right|thumb|250px| An image from inside a Three-torus|3-torus. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe to a small enough observer. This is made more precise in the definition below.