Circle actions on oriented manifolds with 3 fixed points

TL;DR

This paper classifies circle actions on oriented manifolds with exactly three fixed points, showing specific dimension restrictions and characterizing the weights at fixed points in dimension 8.

Contribution

It establishes dimension constraints and characterizes fixed point weights for circle actions with three fixed points, especially relating to quaternionic projective spaces.

Findings

Dimension of M is a multiple of 4 when there are three fixed points.

In 8 dimensions, weights match those of  quaternionic projective space.

No such 12-dimensional manifold exists with three 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 $M$.