Compute explicitly a fundamental group

Question

I want to compute the $\pi_1(X)$ where $$X=\mathbb{R}^2-(([-1,1]\times \{0\})\cup (\{0\}\times [-1,1]))$$ my only tools at the moment are the basic definitions and the fundamental group of a circle, I think this should be the same group $\mathbb{Z}$ but I don't know how to calculate it explicitly.

Answer 1

$X$ is homotopy equivalent to $\mathbb{R}^2-(0,0).$ To see this retract the $(([−1,1]×{0})∪({0}×[−1,1]))$ to the origin. Then consider loops that contain the origin and loops that do not contain the origin.

Answer 2

The space X in questions is homotopy equivalent to $R^{2}\setminus (0,0)$ because you can immage to extend its hole in the two coordinate direction.

Now $\pi_{1}(X)\cong \pi_{1}(R^{2}\setminus (0,0))$ and the last one is $\mathbb{Z}$ because $R^{2}\setminus (0,0)\cong \mathbb{R}\times S^{1}$ (for example see this: Prove that $\Bbb R^2 - \{0\}$ is homeomorphic to $S^1 \times \Bbb R$. ).

Finally $\pi_{1}(\mathbb{R}\times S^{1})\cong \pi_{1}(\mathbb{R})\times \pi_{1}(S^{1}) \cong 0\times \mathbb{Z}$.