Fundamental group of $\mathbb{R}^{3}$ / $ \{\mathbb{S}^1 \cup \{n $ parallel lines passing through $\mathbb{S}^1\} $

Question

Let $H = \{\mathbb{S}^1 \cup \{n $ parallel lines passing through $\mathbb{S}^1\} $ . The lines are perpendicular to $\mathbb{S}^1$.

I want to calculate the fundamental group of $\mathbb{R}^{3} - H$. Any hint ?

Answer 1

Hint: wlog suppose all the lines are vertical and within a small epsilon of the $z$ axis, and $S^1=\{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2=1,z=0\}$. Let $U$ be $\{(x,y,z)\in\mathbb{R}^3\setminus H\mid x^2+y^2> 2\epsilon^2 \}$ and let $V$ be the rest of $H$ (take a small open thickening so that $U$ and $V$ have non-empty intersection). Now use Van-Kampen's theorem.

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I'll add a little detail seeing as this hint didn't seem to be enough.

Note in the above that $U$ looks like $\mathbb{R}^3$ with a solid cylinder removed and a circle removed (with the cylinder going through it). You should be able to see that this is homotopy equivalent to a torus (in fact it deformation retracts onto a toral subspace). Let $\alpha$ be a generator of the longitudinal component of $\pi_1(U)$ and $\beta$ be a generator for the meridinal component.

You will also notice that $V$ is homotopy equivalent to a wedge of $n$ circles and (with a suitable choice of generators for the fundamental group of $V$) We can see that in the intersection, $\beta$ is homotopic to a loop goes round each generator of $\pi_1(V)$ once (that is, if $\pi_1(V)=\langle x_1,\ldots x_n\rangle$ then $\beta$ is homotopic as a loop to the element $x_1x_2\ldots x_n$. I'll leave the tricky bit of Van Kampen's to you.