Let $S^1$ act smoothly on $m$-dimensional sphere $S^m$ with $m$ even. Let $F\subset S^m$ be the fixed point set. I've learned in class that, by the Lefschetz fixed point formula, we have $\chi(F)=\chi(S^m)=2$ where $\chi$ is the Euler characteristic. But I can't see how the Lefscehtz fixed point theorem (https://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem) implies this result. Is the Lefschetz fixed point formula different to the Lefschetz fixed point theorem? Or is there another way to show this result?
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user302934
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1In the wiki page, the theorem "is a formula". What is Lefschetz fixed point theorem you are referring to? – Arctic Char Sep 04 '21 at 11:42
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@ArcticChar I think the one in wiki page. – user302934 Sep 04 '21 at 11:50
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3A proof can be found here – Arctic Char Sep 04 '21 at 13:55