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Is there an intuitive reasoning behind the relation of topological Euler characteristic and Betti numbers - the former being an alternating sum of the latter (starting with a plus sign at that)? Or is it purely coincidental?

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I like to this about it this way : the Euler characteristic is counting the zeroes of a "generic" vector field $v$, this is the Poincaré-Hopf theorem. On the other hand, integrating this vector field gives a diffeomorphism $f$ isotopic to identity, and fixed point of this diffeomorphism correspond exactly to the zeroes of this vector field. But by Lefschetz fixed point theorem, one has $\#\text{Fix}(f) = \sum (-1)^i tr (f_* : H^i(X) \to H^i(X))$.

So we have $$\chi(X) = \# \{ \text{zero of the vector field } v \} = \# \{ \text{fixed point of } f \} = \sum (-1)^i \text{tr} (f_* : H^i(X) \to H^i(X)) = \sum (-1)^i \text{tr} (\text{id}_* : H^i(X) \to H^i(X)) = \sum (-1)^i \dim H^i(X)$$ where I used that $f$ is isotopic to $\text{id}$ so the induced map in homology is the same.

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