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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).

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    $\begingroup$ It is indeed a very standard (and easy) fact; it is usually stated in cohomology, but this is equivalent through Poincaré duality. One reference (among many) is Milnor-Stasheff, Characteristic classes, Thm. 11.11. $\endgroup$
    – abx
    Sep 26, 2015 at 14:42
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    $\begingroup$ For the dual statement, see Example 7.3.9, p. 249 of the notes below. www3.nd.edu/~lnicolae/Lectures.pdf $\endgroup$ Sep 26, 2015 at 23:03
  • $\begingroup$ @abx and Liviu: Thanks; this answers my question. $\endgroup$
    – Ritwik
    Sep 27, 2015 at 3:42

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