I'm trying to understand the following statement, for $M$ a (closed, symplectic) smooth manifold:
Choose a basis $\{e_0,\ldots,e_N\}$ of $H^*(M;\mathbb{Z})/\text{Tor}$, and denote by $g^{\nu\mu}$ the matrix inverse to $g_{\nu\mu} := \int_M e_\nu \smile e_\mu$. Then $\sum_{\nu\mu} e_\nu g^{\nu\mu} e_\mu$ is Poincaré dual to the diagonal of $M \times M$.
I'm not sure how to interpret the integral which defines $g_{\nu\mu}$. The only way I can think to integrate cohomology classes is by interpreting them as differential forms (using De Rham's theorem), but this only makes sense if we make assumptions about the degrees of $e_\nu$ and $e_\mu$.
Moreover, it's not clear to me how any integral becomes a matrix. Perhaps we want to interpret the integral $g_{\nu\mu}$ as a function from the cohomology ring to itself; then it becomes a matrix written with respect to the basis we've chosen. However, I don't know how to fill in the details.