a few problems about  fundamental groups 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?
 A: For $2)$, a torus with a point deleted should deformation retract to a wedge sum of two circles (note a square with an inside point deleted deforms to its edges and use the induced retraction on the quotient space by identifying the edges of the square). So the fundamental group is isomorphic to $\mathbb{Z}*\mathbb{Z}$.
For $3)$, when $n=2$, it should be a free product of $m$ copies of $\mathbb{Z}$ (I don't really know how to make the statement truly rigorous,but the conclusion is true). For $n > 2$, argue by induction to show that deleting m points has the same fundamental group as deleting $m-1$ points. Let $P_1,\dots,P_m$ be the deleted points. $X$ be the space with $\{P_1,\dots,P_{m-1}\}$ points deleted, $U$ be a nbhd of of $P_m$ in $X$, $V=\mathbb{R}^n-\{P_1,\dots,P_m\}$. $U\cap V$ is homeomorphic to $S^{n-1}$(thus having a trivial fundamental group). And use the van Kampen theorem to conclude $\pi_1(V) \cong \pi_1(X)$.  Thus the fundamental group is trivial for $n>3$. (I believe this is an excercise in Hatcher).
