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Let $G$ be an infinite compact Lie group acting on a compact space $X$. Denote $F=F(G,X)=\{x\in X$ : $gx=x$ for all $g\in G\}$.

Show that if $H^*(B_{G_x};\mathbb{Q})=0$ for all $x \notin F$ and $T^1$ is any circle subgroup of $G$, then $F(G,X)=F(T^1,X)$. Here $B_{G_x}$ denotes the classifying space of the isotropy subgroup $G_x=\{g\in G : gx=x\}.$

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  • $\begingroup$ You shouldn't exclude $\{\text{curly braces}$ from MathJax. I changed $F=F(G,X)=${$x\in X$ for all $g\in G$} to $F=F(G,X)=\{x\in X$ for all $g\in G\}$. The former way yields font mismatches and lack of proper horizontal spacing. $\endgroup$
    – Michael Hardy
    Commented Jan 6 at 0:06
  • $\begingroup$ Also, you can write $F=F(G,X)= \{x\in X \text{ for all } g\in G\},$ with the words "for all" included within MathJax. $\endgroup$
    – Michael Hardy
    Commented Jan 6 at 0:07
  • $\begingroup$ @MichaelHardy Thank you for correcting that, I couldn't write that. $\endgroup$
    – Mehmet Onat
    Commented Jan 6 at 11:02

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