4sphere

The 4-sphere, denoted S^4, is the set of all points in five-dimensional Euclidean space at a fixed distance r from the origin. It is the boundary of the 5-dimensional closed ball B^5(r) and forms a smooth, compact 4-dimensional manifold.

Equivalently, S^4_r = { (x1, x2, x3, x4, x5) ∈ R^5 : x1^2 + x2^2 + x3^2 + x4^2 + x5^2 = r^2 }.

Dimension and symmetry: As a 4-manifold, S^4 is homogeneous under the action of the orthogonal group O(5). It is simply connected and has the standard orientation for n ≥ 2. Its topology is that of a sphere in one higher dimension.

Topological invariants: The singular homology groups are H0 ≅ Z, H4 ≅ Z, and Hk ≅ 0 for k = 1, 2, 3. Its cohomology ring is isomorphic to Z in degrees 0 and 4.

Geometry and measure: The surface area of S^4_r is 8π^2 r^4 / 3; the volume of the interior 5-ball B^5(r) is (8/15)π^2 r^5.

In mathematics and physics, 4-spheres appear in differential geometry, topology, and gauge theory, and as a model geometry in certain theories of gravity and cosmology. They also serve as a basic example of a compact, simply connected 4-manifold.