4-manifold
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In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).
Wikidata facts
- Subclass of
- four-dimensional space
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- topic's main category
- Category:4-manifolds
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- math.stackexchange.com/tags/4-manifolds
- maintained by WikiProject
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Encyclopedic overview
19 sectionsContents
- Topological 4-manifolds
- Smooth 4-manifolds
- Special phenomena in 4 dimensions
- Failure of the Whitney trick in dimension 4
- Geometrization in dimension four
- The Four Dimensional Geometries
- Spherical or compact type
- Euclidean type
- Nilpotent type
- Solvable type
- Isomorphisms between solvable geometries
- Hyperbolic type
- Product of hyperbolic planes
- The tangent space of the hyperbolic plane
- Remaining geometries
- See also
- Footnotes
- References
- External links
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).
4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.
Excerpted from Wikipedia’s “4-manifold” article, available under the CC BY-SA 4.0 licence.