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One can find all over the internet that it is well-known (and obvious) that given a fiber bundle $F \to E \to B$, the equality $\chi(E) = \chi(F)\chi(B)$ holds ($\chi$ is the Euler characteristic). This is supposed to be true without any serious (= other than the Euler characteristics of $B$ and $F$ being defined) restrictions on $B$, $F$ and $E$. The thing is, I cannot find any reference for this statement.

What I know (and what I've found proof of) is that if $F \to E \to B$ is an orientable fibration, then this is indeed true. But how do I get rid of the orientability assumption?

I'd really appreciate any reference or a sketch of proof.

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Have you (if you understand french math) checked Serre, Homologie singulère des espaces Fibrés? I'm pretty sure it's in there. EDIT: It is, but unfortunately in the oriented case too. – Ben Oct 11 '12 at 13:38

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