Find the fundamental group of $X$ Let $X$ be the unit square with corners identified. I was thinking about its fundamental group. My strategy was to visualize it as e CW complex with a single $0$-cell, four $1$-cells (i.e a wedge of four circles) and a $2$-cell attached along the $1$-cells. This will create a relation and $\pi_1(X) = \mathbb{Z} *\mathbb{Z} *\mathbb{Z}$.
Is this correct?

Here's another way to see that $\pi_1(X)$ is isomorphic to the free group with three generators. First of all, by thickening the unique $0$-cell in $X$, one see that $X$ is homotopic equivalent to the sphere minus four points. Then using the stereographic projection, this is homeomorphic to $\mathbb R^2$ minus three points. Thus it's fundamental group is $\mathbb Z*\mathbb Z * \mathbb Z$.
