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Reading the "Differential Topology" of V.Guillemin and A.Pollack, i found a definition of the Euler Characteristic different from the other one using the simplicial complex and betti number (ex. for surface $\chi(S) = F- L +V$). That definition said that $\chi(S) = I(\Delta,\Delta)$ where $\Delta$ is the diagonal of $S\times S$, the self-intersection number of $\Delta$. I know that there is a proof of that definition using the differential forms and the property of cohomolgy group, but i ask if there is another proof not using that, using only basic knowledge of differential topology (like in the Guillemin's book). Thanks for any response.

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