I was asked a few "challenge problems". Maybe it's not that hard, but i don't know how to solve them.
1) What's the fundamental group of $R^3 \setminus \{ \{z\text{-axis}\} \cup \{ x^2 + y^2 =1\}\}$?
2) What's the fundamental group of $(S^1 \times S^1) \setminus \{\text{a point}\}$?
I know that the fundamental group of $(S^1 \times S^1)$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$. but take out a point?
3) What's the fundamental group of $R^n \setminus \{m\text{ distinct points}\}$ $(n \ge 2)$?
I have a feeling that I need to use induction on this?