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Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works), $f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic. By the Lefschetz fixed point theorem (or by counting cells of $X$ and $Y$ with the induced cell structure) one obtains

$$\chi(Y)=n \cdot\chi(X)$$

I wonder if there is a sheaf theoretic proof of this statement, or even better a sheaf theoretic generalization of this statement by replacing the constant sheaf by any other local system.

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Doesn't the Leray spectral sequence give this straight away? –  Gunnar Þór Magnússon Jul 12 '13 at 14:44

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up vote 4 down vote accepted

Generalization: Euler characteristic with coefficients in a rank $n$ local system equals $n$ times ordinary Euler characteristic. Then using Leray (as in Gunnar Magnusson's comment) gives the case of a finite covering (and more generally that of a fibre bundle).

The following `sheafy' proof of the generalization should work. Pick an acyclic cover that trivializes the local system and use the associated Cech resolution. An essentially equivalent way would be to do an induction argument using the Mayer-Vietoris distinguished triangle.

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Yes, that works. I was hoping of something involving the 6 functor formalism, but I really didn't think it though. Thanks for the answer. Best Oliver. –  Oliver Straser Jul 13 '13 at 5:42

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