S1×S2

S1 × S2 denotes the Cartesian product of the circle S1 (the unit circle in the complex plane) and the 2-sphere S2 (the set of unit vectors in R3). As a topological space, S1 × S2 is a compact, connected, orientable 3-manifold. It can be viewed as the trivial S2-bundle over S1, or equivalently as S2 × S1, the mapping torus of the identity on S2.

Topologically, S1 × S2 is a simple yet nontrivial example of a product manifold. Its fundamental group is infinite cyclic, π1(S1 × S2) ≅ Z, coming from the S1 factor; the universal cover is R × S2. The space is orientable and has a product-like local structure: locally it looks like a product of an interval with S1 and S2.

The homology and cohomology of S1 × S2 can be computed from the homology of S1 and S2. The homology groups with integer coefficients are H0 ≅ Z, H1 ≅ Z, H2 ≅ Z, and H3 ≅ Z, with all other groups zero. Consequently, the Euler characteristic χ is 0. The cohomology ring satisfies H*(S1 × S2) ≅ H*(S1) ⊗ H*(S2); it is generated by a degree-1 class from S1 and a degree-2 class from S2, with the product of these generators yielding the degree-3 generator.

S1 × S2 serves as a standard example in 3-manifold topology: it is a compact, orientable, non-simply connected 3-manifold with a straightforward product structure, illustrating how product operations affect fundamental groups, homology, and cohomology in low dimensions. Visually, it can be thought of as a continuous family of 2-spheres parameterized by a circular coordinate.