I'm trying to calculate the fundamental group of the following space:
I've been thinking that I should apply Seifert - Van Kampen theorem but I haven't been able to choose some nice open sets $U$ and $V$ because most of the times I end up with something whose intersection is not path connected or doesn't give me anything interesting to calculate the group of the whole space. Which sets should I consider to prove this using this theorem? Is there an easier way to calculate $\pi_1(X)$ in this case?
I don't know covering spaces theory yet so I can't use those results.
P.S. I think that $\pi_1(X)= \mathbb{Z}$ using that if $A \subset X$ is contractible then $X$ and $X / A$ have the same homotopy type and considering as contractible subsets both segments. Unfortunately, I can't use this result in my proof.
If you move the ends of the segments around inside the body, you can show that the space is homotopy equivalent to $T^2\vee S^1\vee S^1$.
If you wanted to make this rigorous, just find a contractible subspace $A$ of the torus which contains the end points of each segment (a tree in the torus with the ends of the line segments at its leaves will do) and then quotient out this subspace to form $X/A$. As $A$ is contractible this does not change the homotopy type, and as $A$ contains the end points of each segment, it's not hard to see that the quotient will be homeomorphic to $T^2\vee S^1\vee S^1$.
Finally, the torus has fundamental group $\mathbb{Z}×\mathbb{Z}$, and the circle has fundamental group $\mathbb{Z}$. The fundamental group of a wedge of finitely many spaces is the free product of the fundamental groups of the individual spaces, so we get $\pi_1\cong \mathbb{Z}^2 \ast \mathbb{Z} \ast \mathbb{Z}$.