Circle actions on oriented manifolds with 3 fixed points

Abstract

Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $\dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $\mathbb{HP}^2$, and show that there is no such 12-dimensional manifold $M$.