Which is the fundamental group of the following space: $S^1 \times S^1/\cong$ where $\cong$ is the equivalence relation identifying two distinct points $p$ and $q$?
First of all, in the comments I ask what happens when at least one of the points, say $P$, stays in the boundary. It has been answered me that it doesn't matter, since I can construct a homeomorphism to another torus in which these points both belong to the interior of the square. I want to know why the homeomorphism does not create problems with the fact that $P$ identifies with some $P'$ in the other side.
I want to use Van Kampen theorem and I draw a picture of the square with the edges labelled by $a,b,a^{-1}$ and $b^{-1}$. I suppose firstly that the two points are in the interior of the square. How I have to take the two open sets needed to use Van Kampen theorem? Can you provide a detailed formal proof, answering point by point to the solution given below.
If you want to do it this way, I'll draw a picture for you:
We'll take $U$ to be the union of the yellow and red regions, and we'll take $V$ to be the union of the blue and red regions. So $U \cap V$ is the red region.
Now, a few sketchy hints:
Try to convince yourself that $U$ deformation retracts on the line segment joining $p$ and $q$, which is really a circle, because $p$ and $q$ are the same point. What is the fundamental group of $U$ then? What does the generator of $\pi_1(U)$ look like on the diagram?
Similarly, you can convince yourself that $V$ deformation retracts onto the "border" of the square, which is the same thing as your $a$ and $b$ circles joined at the "corner-point". So what is $\pi_1(V)$? And what do its generators look like on the diagram?
$U \cap V$ is an annulus, so its fundamental group is $\mathbb Z$. You should try to identity the image of generator of $\pi_1(U \cap V)$ under the maps $$(i_1)_\star : \pi_1(U \cap V) \to \pi_1(U), \ \ \ \ \ \ (i_2)_\star : \pi_1(U \cap V) \to \pi_1(V),$$ where $i_1 : U \cap V \hookrightarrow U$ and $i_2 : U \cap V \hookrightarrow V$ are the natural inclusions.
Finally, apply Seifert-van Kampen.
Feel free to ask more questions if these hints are too cryptic.
By the way, a simpler way to visualise this is to convince yourself that the torus with $p$ and $q$ identified is homotopy equivalent to the torus plus a line segment glued at its endpoints to $p$ and $q$. This is then homotopy equivalent to a torus plus a circle glued at a single point: