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

TL;DR

This paper proves that smooth circle actions with exactly three fixed points can only occur in dimensions 4, 8, and 16, and shows certain manifolds with specific Pontrjagin number conditions have restricted signatures and genera.

Contribution

It establishes the non-existence of non-trivial smooth circle actions with three fixed points on oriented manifolds outside dimensions 4, 8, and 16, under specific Pontrjagin number conditions.

Findings

Circle actions with three fixed points only in dimensions 4, 8, 16

Certain manifolds with Pontrjagin conditions cannot have non-trivial actions

Rational projective planes above dimension 16 admit no such actions

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.