fundamental group of disk minus two points

Question

I have to calculate fundamental group of disk $D^2$ minus two points and then fundamental group of disk $D^2$ minus $k$ points. I know that fundamental group of $D^2$ minus one point is trivial. Can you help me?

Answer 1

As Christoph pointed out the fundamental group of a disk minus one point is not trivial. It is actually generated by the homotopy class of the loop going around the hole, i.e. we have $$\pi_1(D^2\setminus \lbrace x \rbrace,x') \cong \mathbb{Z}.$$ By using von Kampen's theorem you can now get a description of the fundamental group of the disk with several holes.

Answer 2

An intuitive way is as follows: homotope $D^2 -\{p_1, p_2\}$ to $D^2 - (e_1 \cup e_2) $ where $e_1, e_2$ are two balls. Then attach one of the balls to $\partial D^2$ and deformation retract the other to the boundary of the space created by attaching the first ball to $\partial D^2$. The resulting space is a wedge of two circles whose fundamental group is $\mathbb{Z} * \mathbb{Z} $. Inductively, the disc minus $k$ points will have fundamental group $ \mathbb{Z} * \cdots * \mathbb{Z} $ where we have $k$ copies of $\mathbb{Z}$.