I have the following problem: calculate the foundamental group of $X=\mathbb R^3-(\{(x,y,z) | x^2+y^2=1, z=0\}\cup\{(x,y,z)|x=y=0\})$.
I can visualize a deformation that brings $X$ onto a torus. So being the foundamental group invariant under retraction this two spaces have the same $\pi_1$$: \pi_1(X)\cong \pi_1(S^1\times S^1)\cong \mathbb Z\times \mathbb Z. $If I am not wrong, can someone help me to explicit the retraction? I need to write it.
For a diret formula, see this paper (Proposition 2.3) which I will reproduce:
which uses cyllindrical co-ordinates $r,\theta,z$ and excludes the $z$-axis and points of the form $(1,\theta,0)$. In this case, $\rho:=\sqrt{(r-1)^2+z^2}$ is nonvanishing, so the function $$f_t(r,\theta,z)=(\theta,(1-t)r+t(r-1)/2\rho+t,(1-t)z+zt/2\rho) $$
which allegedly has image torus with radius 1/2 whose central circle is the unit one inthe $xy$ plane.
Example 4 here is also very nice. The idea is that $\mathbb R^2 \setminus \{pt\}$ is a circle, so considering a plane swept around the $z$-axis, this is just $S^1 \times S^1$