I need a reference for the following result (which I can prove myself, but my proof is rather ugly and I would prefer to just cite the statement instead or re-proving it):


Let $M$ be a closed smooth manifold and let $S^1$ act smoothly on $M$. Then the set of fixed points of the action, $M^{S^1}$, is again a (not necessarily connected) smooth manifold and $\chi(M^{S^1}) = \chi(M)$.



Smooth circle action, $\chi(M^{S^1}) = \chi(M)$
See [this wonderful blog post by Pawlowski.](http://chromotopy.org/blog/torus-actions-maximal-tori-2)


