I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure) $M$, having a continuous action $M \times S^1 \rightarrow M$, and the number of points fixed by $S^1$ is finite and non-zero? If not, I would also be interested to know if there is an example where the fixed point set is non-empty and each component of the fixed point set is a manifold admitting a smooth structure.
I guess without the fixed point condition(s) one can take $N \times S^1$ where $N$ is the E8 manifold. I have done a search for references online but didn't find anything.
