Fundamental group of sphere

Question

To show that $S^3$ is not diffeomorphic to $S^2 \times S^1$, I'd like to say that their fundamental groups are not the same. So $\pi_1(S^3)= 0 $ but why is $\pi_1(S^2 \times S^1) = Z $ ?

Answer 1

The fundamental group functor respects products in the sense that, for topological spaces $X, Y$, $$\pi_1(X \times Y) \cong \pi_1(X) \times \pi_1(Y).$$

So, if $Y = \mathbb{S}^1$, then $\pi_1 (X \times Y) \not\cong 0$.