= Zero-framed Hopf-link handle decomposition of the sphere product
{title2=$S^2\times S^2=h^0\cup h^2\cup h^2\cup h^4$}
Write each <sphere> as two <disks>, $S^2=D_-^2\cup D_+^2$. In their product, $D_-^2\times D_-^2$ is a <zero-handle>; $D_+^2\times D_-^2$ and $D_-^2\times D_+^2$ are <two-handles>. Their attaching circles are the core circles of the complementary solid tori in $\partial(D^2\times D^2)$, hence a <Hopf link>. Constant normal directions from the disk factors give both components the zero <Seifert framing>. The remaining $D_+^2\times D_+^2$ is a four-handle. This product construction establishes the actual <diffeomorphism> type, rather than relying on its <intersection form> alone. The resulting intersection matrix is $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$.
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