Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?

Is it true that $\pi_1(M)=0$? (sorry this first bit of the question is silly) Is it true that $M$ has to be homeomorphic to a sphere? If not, what kind of manifolds can one get?

Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?

Is it true that $\pi_1(M)=0$? (sorry this first bit of the question is silly since any $S^k$ can appear) Is it true that $M$ has to be homeomorphic to a sphere? If not, what kind of manifolds can one get?

Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?

Is it true that $\pi_1(M)=0$? Is it true that $M$ has to be homeomorphic to a sphere? If not, what kind of manifolds can one get?

Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?

Is it true that $\pi_1(M)=0$? (sorry this first bit of the question is silly) Is it true that $M$ has to be homeomorphic to a sphere? If not, what kind of manifolds can one get?