Here is the question:
Let $P$ be the projective plane and $K$ be the Klein bottle. What are $\tilde{H_*}(K;\mathbb{Z})$ and $\tilde{H^*}(K;\mathbb{Z})$? What is $\tilde{H_{*}}(K \times P; \mathbb{Z})$?
I got the following hints:
1-It is a question about the Universal Coefficient Theorem and Kunneth Theorem.
2- You should know the relation between homology and cohomology of the Klein bottle to solve it.
My questions are:
What is the relation between homology and cohomology in general?
Could anyone give me more hints and more details about the solution, please?
I will help you understand $H_*(K)$ and $H_*(P)$ but the rest of the problem is just an exercise in understanding the statements of UTC and Künneth so I leave that to you.
Recall that you can construct $P = \mathbb{RP}^2$ by attaching a disk $D^2$ to the circle via a map $S^1 \to S^1$ of degree $2$. From the cellular chain complex you can see that $H_1(P) \cong \mathbb{Z}/2$ and $H_2(P) = 0$.
Recall also that we typically construct $K$ as the quotient of a square where the sides are identified using the scheme $abab^{-1}$. This is equivalent to attaching a disk to a wedge of two circles via the same scheme, and the cellular chain complex looks like
$$ \dots \to 0 \to \mathbb{Z} \stackrel{\partial}{\to} \mathbb{Z} \oplus \mathbb{Z} \stackrel{0}{\to} \mathbb{Z} $$ with $\partial(d) = a + b + a + (-b) = 2a$, where $d$ represents the $2$-cell and $a,b$ represent the $1$-cells as above. Then $H_1(K) \cong \mathbb{Z}/2\oplus \mathbb{Z}$ and $H_2(K) =0$.
Now given these computations you should be able to use UCT and Künneth to finish the problem.