# Computing some fundamental groups [duplicate]

I'm studying algebraic topology and got stuck.

a. $X_n\in \mathbb{R}^3$ is the union of $n$ distinct lines through the origin. Find $\pi_1(\mathbb{R}^3-X_n)$ for each $n$.

b. Let $X$ be the sum of two tori $S_1\times S_1$ by identifying a circle $S_1\times x_0$ in a torus with $S_1\times x_0$ of the other torus. Find $\pi_1(X)$.

For (a), $\pi_1(\mathbb{R}^3-X_1)=\mathbb{Z}.$ But I cannot compute $n\ge 2$ cases.. for (b), I guess the group is $<a,b,c|ab=ba,bc=cb>$ from Van Kampen. Is it correct?

## marked as duplicate by Najib Idrissi, Daniel Fischer♦Jul 14 '15 at 14:46

• $a$ is the same as a sphere with $2n$ holes. – Michael Burr Jul 14 '15 at 12:42
• For (b), you should notice that a cross section of $X$ looks like a figure 8 (you should imagine $X$ as being two doughnuts lying one on top of the other), so $X$ is homeomorphic to $(S^1 \vee S^1)\times S^1$. Of course, it's always a good idea to get practice using Van Kampen's theorem so definitely use that method too to do the calculation. – Dan Rust Jul 14 '15 at 12:43