let $X$ be a CW-complex on which a finite group $G$ acts. define
$$F=\{ (x,gx)\;|\; x\in X ,g \in G \}$$ i want to compute the Euler characteristic of $F$. I wrote $$F=\cup_{g\in G}{F_g}\;\;,\;\; F_g=\{ (x,gx)\;|\; x\in X \}$$ and noted that for a given $g\in G$, $F_g$ is a copy of $X$. In particular, when the action is free, $F$ is a disjoint union of copies of $F_g$ and then $\chi(F)=|G| \chi(X)$, but when the action is not free the intersections $F_g\cap F_h$ are not clear. Thank you for your help.
