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I'm not entirely sure what exactly your left side means, but I'll try for the closest fit. If $M$ is compact oriented and Riemannian, the Hodge theorem gives that the space of $C^\infty$ $i$-forms decomposes as an orthogonal direct sum $(\text{harmonic forms})\oplus im(d)\oplus im(d^*)$. This will imply that if you consider the $d+d^*$ as a map from the space of even degree forms to odd degree forms, then the index is exactly the Euler characteristic on the right side of your equation. I don't know about an online ref. but lots of books (Griffiths-Harris, de Rham, Warner, Wells...) give a proof of the Hodge theorem.