If $\Sigma_1$ and $\Sigma_2$ are surfaces (i.e. compact, oriented 2-manifolds without boundary), is the signature $\tau (\Sigma_1 \times \Sigma_2)$ well-known? Recall that the signature is the difference $b_{2}^{+}-b_{2}^{-}$, where $b_{2}^{+}$ and $b_{2}^{-}$ are the number of $+1's$ and $-1's$, respectively, which are associated to the diagonalized quadratic form $Q_X(a,b)=\langle a \cup b, [M]\rangle$ defined on $H^2(M^4)\times H^2(M^4).$
I have a way of computing this signature for such a $4$-manifold and found that it is always $0$. I was not able to confirm this with any independent online source or publication, and I am hoping someone will be able to confirm or challenge this result.
The signature is $0$. One way to see this is that $\Sigma_1$ is a boundary, so $\Sigma_1 \times \Sigma_2$ is also a boundary, and the signature vanishes on boundaries. You can also directly compute the intersection form using the Kunneth formula.