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I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below:

table

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 $\pi(Klein)=\mathbb{Z} \times \mathbb{Z_2}$?

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  • $\begingroup$ Isn't the fundamental group of the Klein bottle $\Bbb Z\rtimes \Bbb Z_2$ (not $\Bbb Z\times \Bbb Z_2$)? $\endgroup$
    – user228113
    May 24, 2016 at 23:56
  • $\begingroup$ I am not sure of the difference in notation to be honest $\endgroup$
    – amiz9
    May 24, 2016 at 23:57
  • $\begingroup$ $\Bbb Z\rtimes \Bbb Z_2$ is a semi-direct product. In this case, I mean it with operation $$(a,\,b)\cdot (c,\,d)=(a+(-1)^bc\,,b+d)$$ For instance, $\Bbb Z\rtimes \Bbb Z_2$ is not abelian. $\endgroup$
    – user228113
    May 25, 2016 at 0:01
  • $\begingroup$ Also, the table you linked says that $\pi(\text{figure }\infty)=\Bbb Z*\Bbb Z$, not $\Bbb Z\times\Bbb Z$. $H_1$ and $\pi_1$ are two different things: $H_1$ is the first homology group, while $\pi_1$ is the first homotopy group (aka fundamental group). They are closely related, since $H_1$ is the abelianisation of $\pi_1$, but not the same thing. $\endgroup$
    – user228113
    May 25, 2016 at 0:06
  • $\begingroup$ OK thanks. So I am only concerned with $\pi$ right now $\endgroup$
    – amiz9
    May 25, 2016 at 0:09

1 Answer 1

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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".

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  • $\begingroup$ Can we say that a sphere with $n$ holes in has fundamental group $\underbrace{\mathbb{Z} * \cdots * \mathbb{Z}}_{\text{$n$ times}}$? This gives us the fundamental group of the sphere and the torus $\endgroup$
    – amiz9
    May 25, 2016 at 0:11
  • $\begingroup$ Is there any way to think of the Klein bottle in a similar way? $\endgroup$
    – amiz9
    May 25, 2016 at 0:13
  • $\begingroup$ You can learn the general tool for the $\pi_1$ theorem I wrote, and for the special calculations you ask in your comments, by learning about Van Kampen's Theorem, and/or by learning about how to write a presentation of the fundamental group of a CW complex. $\endgroup$
    – Lee Mosher
    May 25, 2016 at 1:49

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