3-sphere

The 3-sphere is the set of points in four-dimensional Euclidean space at a fixed distance from a center. It is the boundary of a 4-ball and is a compact three-dimensional manifold.

Sphere: A sphere is the set of points at a fixed distance from a center in a space of one higher dimension. The 3-sphere extends the familiar sphere by one ambient dimension.
Unit sphere: A unit sphere is a sphere of radius one centered at the origin. Setting the radius to one gives the standard equation for the 3-sphere.
2-sphere: The 2-sphere is the set of points at a fixed distance from a center in three-dimensional space. It is the familiar two-dimensional counterpart, embedded one dimension lower.
Poincaré conjecture: The Poincaré conjecture states that every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. It characterizes the 3-sphere through a striking condition on three-dimensional spaces.
Hopf invariant: The Hopf invariant is an integer associated with certain maps between spheres, measuring linking among their fibers. The Hopf map from the 3-sphere to the 2-sphere has invariant one.
Hypersphere: A hypersphere is the set of points at a fixed distance from a center in a higher-dimensional Euclidean space. The 3-sphere is the four-dimensional case of this general construction.
Spherical coordinates: Spherical coordinates locate points using radial distance and angular parameters. Three angular parameters describe points on a 3-sphere after fixing the radius.
4-ball: A 4-ball is the set of points in four-dimensional Euclidean space at most a fixed distance from a center. The 3-sphere is its boundary, not its interior.
Geometrization conjecture: The geometrization conjecture classifies compact 3-manifolds by decomposing them into pieces with standard geometries. Spherical geometry is one of the geometries used to classify 3-manifolds.
Alexander horned sphere: The Alexander horned sphere is a wildly embedded 2-sphere in three-dimensional space whose complement is not simply connected. It illustrates how an embedded sphere’s topology can differ from its surrounding space.

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3-manifoldBroader topic: It is the simplest closed 3-manifold and the subject of the Poincaré conjecture.