circle actions. They correspond to fake homotopy projective spaces. –  Igor Belegradek Mar 18 2010 at 12:57 1 One example: The sphere $S^{n+m+1}$ is the join $S^n*S^m$of $S^n$ and $S^m$, so having $S^1$ act on each of the factors you get an action on $S^{n+m+1}$. link | flag edited Mar 18