I need to compute the fundamental group of $X:=\Bbb{R}^3-\Bbb{R}\times\{(0,0)\}$. In previes question I proved that $A:=\{0\}\times S^1$ is a retraction of $X$ so I think I need to use it somehow but I'm not sure how. I know that if A is a retract of X, the homomorphism of fundamental groups induced by inclusion j : A → X is injective.
I will appreciate any idea
Let $Y=\{(0,y,z)-(0,0,0)\}$. Consider $H_t(x,y,z)=(tx,y,z), t\in[0,1]$ restricted to $X$, $H_1=Id_X$, the restriction of $H_t$ to $Y$ is the identity, and $H_0(X)\subset Y$. This implies that $Y=\mathbb{R}^2-\{0\}=S^1\times \mathbb{R}_+^*$ (the last identification is defined by the map $(x,y)\rightarrow ({x\over{\|x\|}},\|x\|)$) is a deformation retract of $X$, we deduce that $\pi_1(X)=\mathbb{Z}$.