I need to calculate $\pi_1 (T \setminus \{P_1, P_2\})$ where $T$ is the two-dimensional torus, and $P_1, P_2$ are any two points. So I need to calculate the fundamental group of the twice punctured torus. I don't even know where to begin with this.
[Math] Fundamental group of twice punctured torus
algebraic-topologygeneral-topology
Related Solutions
You can use van Kampen's Theorem. The upshot is that you get a semi-direct product:
$\pi_1M_f\cong\pi_1X\rtimes_{f_*}\mathbb{Z}$.
Here is the fundamental polygon of the torus.
The punctured torus simply has a point $P$ removed. It's best to think of $P$ having been removed from the gray area. Your proof claims any lift of a generator of $\pi_1(U\cap V)$ is a loop (with winding number 1) around $P$. Recalling that elements of fundamental groups are invariant under homotopies, we can retract this loop to run around the boundary. Starting at the bottom right of our diagram and traveling counterclockwise, we see this look $ABA^{-1}B^{-1}$.
Best Answer
First note that $T\setminus \{P_1, P_2\}$ deformation retracts to the union of three copies of $S^1$ touching pairwise, which is homeomorphic to $\bigvee^3_{i=1}S^1$.
So $T \setminus \{P_1, P_2\}$ is homotopy equivalent to $\bigvee^3_{i=1}S^1$, which implies that $$\pi_1(T \setminus\{P_1, P_2\}) \cong \pi_1\Bigg(\bigvee^3_{i=1}S^1\Bigg) \cong \mathbb{Z} * \mathbb{Z} * \mathbb{Z}.$$ (You can prove the isomorphism via Van Kampen's Theorem.)