Circle actions on six dimensional oriented manifolds with isolated fixed points

TL;DR

This paper classifies circle actions on 6-dimensional oriented manifolds with isolated fixed points by analyzing fixed point data and shows how to reduce it to an empty set through equivariant connected sums, leading to fixed-point-free actions.

Contribution

It introduces a method to classify fixed point data of circle actions on 6-manifolds and demonstrates how to systematically eliminate fixed points via equivariant connected sums.

Findings

Fixed point data can be reduced to empty by specific operations.

Successive equivariant connected sums lead to fixed-point-free actions.

The classification applies to actions with isolated fixed points on 6-manifolds.

Abstract

To classify a group action on a manifold, the data associated with the fixed point set is essential. In this paper, we classify the fixed point data of a circle action on a 6-dimensional compact connected oriented manifold with isolated fixed points, where the fixed point data consists of the collection of signs and weights at the fixed points. We show that this fixed point data can be reduced to the empty collection by performing a sequence of operations. Specifically, we prove that one can successively take equivariant connected sums at fixed points with $S^6$, $\mathbb{CP}^3$, or 6-dimensional analogues of the Hirzebruch surfaces (and their oppositely oriented counterparts), resulting in a fixed-point-free action on a compact connected oriented 6-manifold.