IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$?

Question

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$?

$\pi(X)$ denotes the fundamental group of $X$.

Answer 1

Viewing the torus as a square with opposite edges identified in the usual way. If you take out a point, this becomes a punctured square. You may deformation retract the interior of the square to its boundary. What happens if you identify just the edges of the square in the manner of constructing the torus? What space to you get?

Now suppose you remove two points. Draw a vertical line down the center of the square. This line separates the square into two halves. Without loss of generality, there is one removed point in the right half, and one in the left half. This deformation retracts to the boundary of the square along with the vertical line. What space does this become when edges are identified?