I'm trying to prove that the fundamental group of the torus is $\Bbb Z \times \Bbb Z$, and I've been checking different questions on this site, however in many of the questions people use the Seifert-van Kampen theorem, which we haven't study in my class, and from what I've read here and here, I don't understand completely.
I've figure, that since $\Pi_1(S^1)=\Bbb Z$ and the torus is the product of two circles, then $\Pi_1(T)= \Pi_1(S^1)\times \Pi_1(S^1)=\Bbb Z \times \Bbb Z$, but I don't think this is valid.
The book I'm using is General Topology by S. Willard, and the exercise is $34B.3$.
Best Answer
Yes, this is valid: the functor $\pi_1$ preserves the cartesian product.
Assume we have a loop $\gamma:(x,y)\leadsto (x,y)$ in the product space $X\times Y$, i.e. $\gamma$ is continuous $[0,1]\to X\times Y$ with $\gamma(0)=\gamma(1)=(x,y)$. Then $\gamma$ is uniquely determined by its coordinate functions $\gamma_X$ and $\gamma_Y$ (assuming $\gamma(t)=(\gamma_X(t),\gamma_Y(t))$ for all $t$) which are loops in $X$ and $Y$, respectively, and this induces an isomorphism $$\pi_1\left(X\times Y,\,(x,y)\right)\ \cong\ \pi_1(X,x)\,\times\,\pi_1(Y,y)\,.$$