I’m having trouble solving this exercise.
Let $C=[0,1]^3$ be the standard cube in $\mathbb{R}^3$ And let $X$ be the edges. Now I should compute the fundamental group of $Y$ where $Y$ is obtained collapsing the 3 edges of $X$ with an extremity in the origin to a point.
I tried to use van Kampen but I don’t know which open set consider.
We have a homotopy $H_t([(x,y,z)])=[(tx,ty,tz)]$ between $\operatorname{id}_Y$ and $pt$, i.e., $Y$ is contractible. So $\pi_1(Y)$ is trivial.
Edit: OP clarified it was $X/\sim$ rather than $C/\sim$. In this case, $\pi_1(X/\sim)=\pi_1(X)=\pi_1(S^2-\text{ 6 points})$ is the free group on 5 generators.