How to compute the Euler characteristic of $S^2\times S^2$?

Question

How to compute the Euler characteristic of $S^2\times S^2$ ?

Answer 1

Use $$\chi\left(\mathbb{S}^2\right)=2$$ and $$\chi(A\times B)=\chi(A)\cdot\chi(B)$$

Answer 2

Use cellular decomposition, or a multiplicativity of Euler characteristic, or compute homology and take alternating sum of Betti numbers.

Answer 3

The most general and most elegant way is: let $p_X$ be the Poincaré polynomial of $X$; then $p_X(-1)=\chi(X)$. Now $p_{X\times Y}=p_X p_Y$ (Künneth Theorem).