Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion). Now consider the diagonal $\Delta_{M}$ inside $M\times M$. Suppose we write it as $$ \Delta_{M} = \sum g^{ij} \mu_i \times \mu_j \in H_*(M \times M, \mathbb{Z}).$$ My question is, what are these $g^{ij}$? Based on reading one paper, it seems to me the following ought to be true (but I do not actually see why this is so):
Let $$ g_{ij} := \mu_i \cdot \mu_j, $$ where $\cdot$ is the topological intersection. And then it seems to me $$ g^{ij} := (g^{-1})_{ij}. $$ Is this some standard/obvious fact? The paper I am referring to is "Mathematical theory of Quantum Cohomology" by Ruan and Tian
https://projecteuclid.org/download/pdf_1/euclid.jdg/1214457234
End of page 262 and beginning of page 263 has the relevant discussion (I am using a slightly different notation from them).