Why are $\Bbb S^4$ and $\Bbb S^2×\Bbb S^2$ not homeomorphic

Question

I am stuck in proving that $\mathbb{S}^4$ and $\mathbb{S}^2\times\mathbb{S}^2$ are non-homeomorphic.

I have thought to compute the fundamental group of $\mathbb{S}^4$ minus a point (which is trivial). But I am not able to compute the fundamental group of $\mathbb{S}^2\times\mathbb{S}^2$ with a point removed.

Can anyone give me a hint?

Answer 1

This is easy if one has the apparatus of homology theory to hand: $H_2(X)$ is zero when $X=S^4$ but nonzero when $X=S^2\times S^2$. An alternative is to use the second homotopy group $\pi_2$ instead.

I don't know of a proof that just involves the fundamental group.