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I'm trying to calculate the fundamental group of a surface using (i) deformation retracts and (ii) Van Kampen's Theorem. I'm really struggling understanding the group theory behind it and the interactions behind the different fundamental groups involved ($\pi(U), \pi(V),$ and $\pi(U\cap V)$). I would really appreciate it if someone could help me understand this.

I am trying to calculate the fundamental group of the following figure:

enter image description here

The deformation retract of this figure is shown below. Using Van Kampen's Theorem then, I choose $U$ to be the open region consisting of one triangle and a portion of the line segment connecting the two (the green region in the figure below), and similarly for $V$ (the red region in the figure below).

enter image description here

$\pi(U\cap V)$ is trivial, as $U \cap V$ may be deformed to a point which has a trivial fundamental group. For $U$ and $V$, either of these may be continuously deformed to a shape with fundamental group isomorphic to the integers, or the free group on $1$ generator. It's not particularly clear to me how this helps us find $\pi(X)$, and how the relations on each fundamental group can be translated to relations on $\pi(X)$.

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So, $\pi_1(U)$ is generated by one element say $a$ and $\pi_1(V)$ is generated by the element $b$, now since $\pi_1(U\cap V)=0$ this means there are no more relations between $a$,$b$. So the group that you get for $\pi_1(U\cup V)$ is just the group generated by $(a,b)$ with no relations, i.e. the free group generated by two elements $\mathbb{Z}*\mathbb{Z}$.

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  • $\begingroup$ Deceptively simple - thank you! $\endgroup$ – user312437 Apr 18 at 15:06

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