 Intersection form of a product of two closed oriented surfaces

Here $H$ is the hyperbolic form with matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. The degree-two cohomology consists of the two factor orientation classes and $4gh$ products of degree-one classes. The factor classes pair once and have zero squares. In symplectic bases $a_i,b_i$ and $c_j,d_j$, $(a_i{c}_j)(b_k{d}_l)=-{\delta }_{ik}{\delta }_{jl}{\omega }_1{\omega }_2$ and $(a_i{d}_j)(b_k{c}_l)={\delta }_{ik}{\delta }_{jl}{\omega }_1{\omega }_2$. Each pair $(i,j)$ contributes two hyperbolic forms; the fiber classes contribute one more. The sign in the first formula comes from interchanging two odd-degree factors.