I am trying to solve Exercise 14.37 on the differential-form book by Bott and Tu:
Let $\pi:X\rightarrow Y$ be any map and $\{U\}$ a finite good cover of $Y$. Show that the Euler characteristics of $X$ \begin{eqnarray} \chi(X)=\sum_{p,q}\sum_{\alpha_{_0}<\cdots<\alpha_p}(-1)^{p+q}\text{dim }H^q(\pi^{-1}U_{\alpha_{_0}\cdots\alpha_p}). \end{eqnarray}
It seems that the proof would follow if the spectral sequence stops at $E_2$, but it cannot be the case in general. I guess that the ''finite good cover'' should have been used somewhere.