Product of 2-sphere and circle

Definition
This topological space is defined as the Cartesian product, equipped with the product topology, of the 2-sphere $$S^2$$ and the circle $$S^1$$. It is denoted $$S^2 \times S^1$$ or $$S^1 \times S^2$$.

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

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

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

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

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

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

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

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