S2×S2

S^2×S^2 denotes the Cartesian product of two 2-spheres, forming a smooth 4-manifold. It is diffeomorphic to CP^1×CP^1 and, in the Hirzebruch surface classification, corresponds to F0. It can also be realized as the smooth quadric surface Q in complex projective 3-space CP^3, viewed as a complex manifold.

Topologically, S^2×S^2 is compact and oriented, with fundamental group trivial. Its Euler characteristic is χ = χ(S^2)χ(S^2) = 4. The homology groups are H0 ≅ Z, H2 ≅ Z⊕Z, H4 ≅ Z, with other homology groups vanishing. The cohomology ring is generated in degree 2 by the pullbacks a and b of the orientation classes from each S^2 factor, subject to a^2 = 0 and b^2 = 0, while a∪b generates H^4 ≅ Z.

The intersection form on H^2 is the hyperbolic form with matrix [[0,1],[1,0]] relative to the basis (a,b); it has signature zero and b2 = 2, with one positive and one negative eigenvalue. Consequently, the manifold is spin, since w2 = 0, and has a vanishing signature.

Geometrically, S^2×S^2 carries a natural complex structure, making it a complex surface with Hodge numbers h^{1,1} = 2 and h^{p,q} = 0 for other p ≠ q in (0,2). It admits a Kähler (in fact, product) metric and thus a symplectic structure. There are two natural projections to the factors, giving two distinct S^2-fibrations.

In summary, S^2×S^2 is a fundamental example of a simply connected, spin, compact complex surface with b2 = 2 and hyperbolic intersection form, serving as a basic building block in the study of smooth 4-manifolds.