I have been solving some past exam questions and I came across the following question.
Let $X:= S_1 \cup (\mathbb R \times \{0\})\cup (\{0\}\times \mathbb R)$ be the union of the unit circle, the $X$-axis and the $Y$-axis. Compute the fundamental group of $X$ in terms of generators and relations.
I am not really sure how to go about this. My guess is that if I could find a retract of this space whose fundamental group is known then the question is solved. I am also wondering if the Seifert-van Kampen theorem would work but I have no clue. Any help would greatly be appreciated.
Your space is $X=S^1\cup(\mathbb R\times \{0\})\cup(\{0\}\times\mathbb R)$, which is the union of the circle of radius $1$ centered in $(0,0)$ with the $x$ and $y$ axis.
The space $X$ is homotopic to $S^1\cup (\{0\}\times I)\cup(I\times \{0\})$, so a circle with two diameters.
Our space has now the form $\oplus$ and if we identify the two diameters with a point we obtain the wedge sum of four circles, whose fundamental group is the free product on four generators with no relations: $\langle a,b,c,d|\emptyset \rangle$.
Therefore $\pi_1(X)\cong\pi^1(S^1\vee S^1 \vee S^1 \vee S^1)\cong\mathbb Z*\mathbb Z*\mathbb Z*\mathbb Z$.