Circle Actions on Oriented Manifolds With 3 Fixed Points

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 M=8, then the weights at the fixed points agree with those of an action on the quaternionic projective space HP², and show that there is no such 12-dimensional manifold M.