Product of 3-sphere and real projective plane

Definition
This topological space is defined as the Cartesian product (equipped with the product topology) of the defining ingredient::3-sphere $$S^3$$ and the defining ingredient::real projective plane $$\R\mathbb{P}^2$$. It is denoted $$S^3 \times \R\mathbb{P}^2$$ or $$\R\mathbb{P}^2 \times S^3$$.

Interesting feature
This topological space is one of the simplest examples of a connected manifold such that there exists another connected manifold, namely product of real projective three-dimensional space and 2-sphere, such that both manifolds have the property that their corresponding homotopy groups are isomorphic to each other but the manifolds themselves are not homotopy-equivalent spaces.

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

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

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

$$H^p(S^3 \times \R\mathbb{P}^2; \mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & p = 0,3 \\ \mathbb{Z}/2\mathbb{Z}, & p = 2,5 \\ 0, & p = 1,4, p \ge 6 \\\end{array}$$