Twosphere

Twosphere is a term used in mathematics to denote the Cartesian product of two 2-spheres, written as S^2 × S^2. As a topological space, it is a compact, orientable four-dimensional manifold that carries a natural product structure.

Construction and geometry: Twosphere can be formed by pairing points from each sphere, with the standard round metric on each factor leading to a product metric on the four-dimensional space. In the language of complex geometry, the underlying smooth manifold can be viewed as CP^1 × CP^1, since CP^1 is diffeomorphic to S^2. This makes twosphere a simple example of a complex surface that is also a toric variety.

Topological and geometric properties: Twosphere is simply connected, and its second cohomology group is H^2(twosphere) ≅ Z ⊕ Z, reflecting the two independent 2-dimensional classes coming from each factor. The Euler characteristic is χ(twosphere) = χ(S^2) χ(S^2) = 4. These properties make twosphere a standard test case in four-manifold theory and in the study of surface bundles and product spaces.

Context in geometry and physics: In complex geometry, the same manifold is described as CP^1 × CP^1, a well-known example of a Kähler, projective surface. The product structure also provides a convenient setting for illustrating product metrics, curvature calculations, and various toric and algebro-geometric constructions. In physics, twosphere serves as a compact, orientable model space for exploring ideas in higher-dimensional geometry and field theories.

See also: S^2, CP^1, CP^1 × CP^1, S^2 × S^2, product manifold, Hirzebruch surfaces.