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Let $(Z,A)$ a compact ENR pair, then $$\chi(Z)=\chi_c(Z-A)+\chi(A)$$ where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean neighborhood retract). According to the book Transformation Groups by Tammo tom Dieck, page 230, proposition 1.12 $$\chi(X^{K}/NK)=\displaystyle\sum_{(H_i)}{\chi_c(X^{K}_{(H_i)}/NK)}$$ where $X$ is a compact $G$-ENR, with $G$ a compact Lie group, $K$ is a (closed?) subgroup of $G$ and $H_i$'s run through a complete set of conjugacy classes of (closed?) subgroups of $G$. Each $X^{K}_{(H_i)}/NK$ is supposed to be a subtraction $X_i-Y_i$ where $(X_i,Y_i)$ is a compact ENR pair, but I do not see what that spaces are. Any suggestion?, please. Do $K$ and each $H_i$ need to be closed?.

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