Using Van-Kampen in $S^1\times S^1$ as cylinder I was trying to use Van-Kampen theorem to compute $S^1 \times S^1$ using its representation as a cylinder.
$U$= from a little below the middle circle to the upper part of the cylinder. 
$V$= from a little above the middle circle to the bottom part of the cylinder.
Then $U$ is homotopic to the upper $S^1$. 
Similarly, $V$ is homotopic to the bottom $S^1$
The intersection of $U$ and $V$ is homotopic to the middle circle, $U \cap V = S^1$.
Then $\pi(U)=<\alpha>$, $\pi(V)=<\beta>$, $\pi(U\cap V)=<\gamma>$.
Now $\alpha$ and $\beta$ are homotopic since the upper circle and the bottom circle are relative.
$\pi(X)=<\alpha,\beta \mid \alpha=\beta>=<\alpha>$.
I know this is wrong but I don't understand where. I know that using the rectangular representation of the torus we can use Van-Kampen, just taking a point out and then a circle and we will get $\pi(X)=<\alpha,\beta \mid \alpha*\beta*\alpha^{-1}*\beta^{-1}=1>$. 
I am trying to use van Kampen in this example and I cant do it.
Even more general, Can we use van Kampen in $X\times Y$.

The product $S^1\times S^1$ is not a cylinder. It's a torus.
So what you've done is a correct computation of the fundamental group, but of a wrong space.
In general if you want to compute the fundamental group of a product of spaces, you don't need van Kampen. The group $\pi_1(A\times B)$ is just going to be the product of the fundamental groups $\pi_1(A)\times\pi_1(B)$. It's easy to see, since any loop in $A\times B$ is just a product of loops in $A$ and in $B$.
