Circle actions on 6-dimensional oriented manifolds with 4 fixed points

TL;DR

This paper classifies circle actions on 6-dimensional oriented manifolds with four fixed points, showing they resemble rotations on spheres or linear actions on complex projective 3-space, including exotic cases.

Contribution

It provides a complete classification of fixed point data for such circle actions, linking them to known geometric models.

Findings

Fixed point data matches disjoint unions of sphere rotations or linear actions on P^3

Includes Petrie's exotic P^3 action as a special case

Classifies all possible fixed point configurations under the given conditions

Abstract

In this paper, we classify the fixed point data (weights and signs at the fixed points), of a circle action on a 6-dimensional compact oriented manifold with 4 fixed points. We prove that it agrees with that of a disjoint union of rotations on two 6-spheres, or that of a linear action on $\mathbb{CP}^3$. The former case includes that of Petrie's exotic action on $\mathbb{CP}^3$.