I want to prove that Euler characteristic of the product of two compact oriented manifolds is the product of their Euler characteristics.
As always I do, I'm considering Guillemin-Pollack definitions, i.e., the Euler characteristic of M, compact and oriented, $\chi(M) = I(\Delta,\Delta)$ where $\Delta$ is the diagonal of $M\times M$ and $I(\Delta,\Delta) = I(i,\Delta) =$ sum of orientation numbers of each $p\in i^{-1}(\Delta)$ using pre image orientation. Here $i:\Delta \to M \times M$ is the inclusion.
Help!