Is much known about $S^1$-actions on the following simple spaces?:
1) $D^2$ the disk
2) More generally $D^n$ the n-ball
3) $S^1$ the circle
In particular, does every $S^1$-action on the disk (or general n-ball) have a fixed point? I.e. there is an $x_0\in D^2$ (or $D^n$) such that for all $g\in S^1$ we have $gx_0=x_0$.
Are the general actions just rotations about the origin? I believe $S^1$ must at least send the boundary to itself, because the $\epsilon$-neighborhood of a boundary point is different than the $\epsilon$-neighborhood of an interior point. And is it just rotations for $S^1$ on the circle?