Product of two 2-spheres

Definition
This topological space is defined as the Cartesian product of two copies of the 2-sphere $$S^2$$, equipped with the product topology. It is denoted $$S^2 \times S^2$$.

Homology groups
The homology groups with coefficients in integers are as follows:

$$H_p(S^2 \times S^2;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad p = 0,4 \\ \mathbb{Z}^2, & \qquad p = 2 \\ 0, & \qquad p = 1, p = 3, p \ge 4 \\\end{array}$$

The homology groups with coefficients in any module $$M$$ over a ring are as follows:

$$H_p(S^2 \times S^2;\mathbb{Z}) = \lbrace\begin{array}{rl} M, & \qquad p = 0,4 \\ M^2, & \qquad p = 2 \\ 0, & \qquad p = 1, p = 3, p \ge 4 \\\end{array}$$

Cohomology groups
The cohomology groups with coefficients in integers are as follows:

$$H^p(S^2 \times S^2;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad p = 0,4 \\ \mathbb{Z}^2, & \qquad p = 2 \\ 0, & \qquad p = 1, p = 3, p \ge 4 \\\end{array}$$

The homology groups with coefficients in any module $$M$$ over a ring are as follows:

$$H^p(S^2 \times S^2;\mathbb{Z}) = \lbrace\begin{array}{rl} M, & \qquad p = 0,4 \\ M^2, & \qquad p = 2 \\ 0, & \qquad p = 1, p = 3, p \ge 4 \\\end{array}$$