I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below:
Why is this the case?
Is it is because they are all realted to the free group on n generators?
I saw in this question Fundamental group of complement of $n$ lines through the origin in $\mathbb{R}^3$ that the free group on n generators corresponds to spaces which are the complement to lines in the origin.
So the circle relates to the open ball with 1 hole drilled though, $\pi(circle)=\mathbb{Z}$, the figure 8 space with two holes drilled through, $\pi(figure 8)=\mathbb{Z} * \mathbb{Z}$, the sphere with zero holes drilled through, $\pi(sphere)=0$.
How could we relate this to the fundamental group of the Klein bottle $\pi(Klein)=\mathbb{Z} \times \mathbb{Z_2}$?
For $H_1$, there's a theorem that every finitely generated abelian group is a quotient of a finite rank free abelian group $\underbrace{\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}}_{\text{$n$ times}}$ for some $n$ (the theorem is actually more precise than this, but I'm writing it this way to emphasize $\mathbb{Z}$).
For $\pi_1$, there's a theorem that every finitely generated group is a quotient of a finite rank free group $\underbrace{\mathbb{Z} * \cdots * \mathbb{Z}}_{\text{$n$ times}}$ for some $n$. In this case there is no more precise statement in general. But, there are many special ways to study the kernel of the quotient homomorphism; this comes under the banner of "group presentations".