Product of 3-sphere and circle

Definition
This topological space is defined as the Cartesian product of the 3-sphere $$S^3$$ and the circle $$S^1$$, equipped with the product topology. It is denoted $$S^3 \times S^1$$ or $$S^1 \times S^3$$.

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

$$H_p(S^3 \times S^1;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad p = 0,1,3,4 \\ 0, & p = 2, p \ge 5 \\\end{array}$$

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

$$H_p(S^3 \times S^1;M) = \lbrace\begin{array}{rl} M, & \qquad p = 0,1,3,4 \\ 0, & p = 2, p \ge 5 \\\end{array}$$

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

$$H^p(S^3 \times S^1;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad p = 0,1,3,4 \\ 0, & p = 2, p \ge 5 \\\end{array}$$

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

$$H^p(S^3 \times S^1;M) = \lbrace\begin{array}{rl} M, & \qquad p = 0,1,3,4 \\ 0, & p = 2, p \ge 5 \\\end{array}$$