Fundamental group of the complement in $\Bbb R^3$ of the union of the $x$-axis, the $y$-axis, and the cylinder $S^1\times [0,1]$

Question

Let $A$ denote the union of $x$ and $y$ axes in $\Bbb R^3$, and let $B$ denote the cylinder $S^1\times [0,1]$ in $\Bbb R^3$. I am asked to compute the fundamental group of the space $X=\Bbb R^3-(A\cup B)$. It seems that I may take suitable subspaces of $X$ and use the van Kampen theorem, but I have no idea for this. I can't even see a way to compute the fundamental groups of $\Bbb R^3-A$ and $\Bbb R^3-B$. Any helps will be greatly appreciated

Answer 1

Is this "a way to compute" you're looking for?

Let $\approx$ represents isomorphism, $\cong$ homeomorphism, and $\simeq$ homotopy equivalence.

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Calculation:

Let $M=A\cup B$, and $X=\Bbb{R}^3\setminus M$.

Claim:

1. $S^1\times[0,1]\simeq S^1\simeq S^1\times D^2$.

2. $\{(x,0,0)\in\Bbb{R}^3:x\in\Bbb{R}\}\cong \{(0,y,0)\in\Bbb{R}^3:y\in\Bbb{R}\}\simeq[0,1]\simeq\{*\}$

3. The removal of any line is the same as creating a tube.

4. $\Bbb{R}^3\simeq B^3\simeq\{*\}$ and $\Bbb{R}^3\cup_f\{*\}\cong S^3$ where $f$ is the attaching map.

Then, we'll go through the following process:

We can see that the white region represents the tubes in the space while the shaded region is solid and note that we're looking at the projection of a 3-dim space so the shaded region is path connected. Then in fig.2 the four isolated tube can be moved to the same side and form a 4-fold connected solid torus denoted by $T\simeq\bigvee_{i=1}^4S^1\implies\pi_1(T)\approx \Bbb{Z}^{*4}$. The other part is a contains a crossed tube, if we deformation retract the outer outer boundary to nearly the boundary of that tube then we actually get $S^2\setminus\{p_1,p_2,p_3,p_4\}\simeq\bigvee_{i=1}^3S^1\implies\pi_1(S^2\setminus\{p_1,p_2,p_3,p_4\})\approx\Bbb{Z}^{*3}$. Then apply Van-Kampen's Thm, since the intersection is contractible, there is an isomorphism $$i:\pi_1(T)*\pi_1(S^2\setminus\{p_1,p_2,p_3,p_4\})\to\pi_1(X)$$

Thus, $\pi_1(X,x_0)\approx\Bbb{Z}^{*7}$.

The calculation above assumes that $S^1\times[0,1]$ is connected to the $x,y$-axis, otherwise if the cylinder is centered at some other point, say $(100,100,100)$, then the fundamental group of $X$ wouldn't be $\Bbb{Z}^{*7}$ but rather $\Bbb{Z}^{*4}$ (if I computed correctly), thus the position of the cylinder needs to be specified.

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Hope this helps.