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Given a pushout $P=B\cup_AC$ which we represent as a commutative diagram

$$ \begin{matrix} A & \stackrel{f}{\rightarrow} & B\\ \downarrow{g} & &\downarrow{k} \\ C &\stackrel{h}{\rightarrow} & P \end{matrix} $$ the euler characteristic is given by $\chi(P)=\chi(C)+\chi(B)-\chi(A)$.

Do we have a similar situation when a space is constructed from a pullback, i mean what can be said of $\chi(X)$ when a space $X$ is given by a pullback : $$ \begin{matrix} X&\stackrel{f}{\rightarrow}&Y\\ \downarrow{g}&&\downarrow{k}\\ Z&\stackrel{h}{\rightarrow}&T \end{matrix}$$

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    At the best you mean homotopy pullback, since pullback isn't always homotopy invariant. For example, the pullback of $* \to X \gets *$ is $*$, but the pullback of $PX \to X \gets PX$, where $P$ denotes path space is $\Omega X$, the loop space on $X$.2011-12-02
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    if the spaces/maps are very nice then $\chi(X)=\chi(Y)\chi(Z)/\chi(T)$ (I don't specify what "very nice" means, so this is not an answer)2011-12-02

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