4-sphere S^4
E911210
Riemannian manifold
closed manifold
compact manifold
connected manifold
differentiable manifold
homology sphere
n-sphere
oriented manifold
simply connected manifold
smooth manifold
topological space
The 4-sphere S⁴ is the four-dimensional analogue of the ordinary sphere, a compact, smooth, simply connected manifold that serves as a fundamental object in topology and differential geometry.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Riemannian manifold
ⓘ
closed manifold ⓘ compact manifold ⓘ connected manifold ⓘ differentiable manifold ⓘ homology sphere ⓘ n-sphere ⓘ oriented manifold ⓘ simply connected manifold ⓘ smooth manifold ⓘ topological space ⓘ |
| admitsRoundMetric | true ⓘ |
| boundaryOf | 5-ball B^5 ⓘ |
| canBeDescribedAs | SO(5)/SO(4) ⓘ |
| cohomologyRing | Z[α]/(α^2) with deg(α)=4 ⓘ |
| constantSectionalCurvature | 1 ⓘ |
| definedAs | {x in R^5 : ||x|| = 1} ⓘ |
| dimension | 4 ⓘ |
| embeddedIn | R^5 ⓘ |
| EulerCharacteristic | 2 ⓘ |
| fundamentalGroup | trivial group ⓘ |
| generalizes |
2-sphere S^2
ⓘ
3-sphere S^3 ⓘ |
| hasCanonicalRiemannianMetric | round metric ⓘ |
| homologyGroupH_0 | Z ⓘ |
| homologyGroupH_1 | 0 ⓘ |
| homologyGroupH_2 | 0 ⓘ |
| homologyGroupH_3 | 0 ⓘ |
| homologyGroupH_4 | Z ⓘ |
| isBoundary | true ⓘ |
| isCompact | true ⓘ |
| isConnected | true ⓘ |
| isModelSpaceFor | constant positive curvature geometry in dimension 4 ⓘ |
| isOrientable | true ⓘ |
| isParallelizable | false ⓘ |
| isPrototypeOf | compact simply connected 4-manifold ⓘ |
| isSimplyConnected | true ⓘ |
| isSimplyConnectedAtInfinity | true ⓘ |
| isSimplyConnectedHomologySphere | true ⓘ |
| isSymmetricSpace | true ⓘ |
| notation | S^4 ⓘ |
| pi_1 | 0 ⓘ |
| pi_2 | 0 ⓘ |
| pi_3 | Z ⓘ |
| pi_4 | Z_2 ⓘ |
| usedIn |
differential geometry
ⓘ
gauge theory ⓘ general relativity NERFINISHED ⓘ homotopy theory ⓘ topology ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.