Non-existence of circle actions on oriented manifolds with three fixed points except in dimensions 4, 8 and 16

Abstract

Let $M$ be a smooth $4k$-dimensional orientable closed manifold, and assume that $M$ has at most two non-zero Pontrjagin numbers which are associated to the top dimensional Pontrjagin class and the square of the middle dimensional Pontrjagin class. We prove that the signature of $M$ being equal to $1$ and the $\hat{A}$-genus of $M$ vanishing cannot hold at the same time except $\dim M=8, 16$. As an application, we claim that the dimensions of oriented $S^1$-manifolds with exactly three fixed points are only $4, 8$ and $16$, and the rational projective plane whose dimension is greater than $16$ has no smooth non-trivial $S^1$-action.