Product of real projective three-dimensional space and 2-sphere

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

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 3-sphere and real projective plane, 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, something which can easily be seen from the fact that they do not have isomorphic homology groups, or even from the fact that $$\mathbb{R}\mathbb{P}^3 \times S^2$$ is orientable whereas $$S^3 \times \R\mathbb{P}^2$$ is non-orientable.

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

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

These can be computed by combining knowledge of the homology of real projective space, the homology of spheres, and the Kunneth formula.

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

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