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?.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.