Circle actions on six dimensional oriented manifolds with isolated fixed points

TL;DR

This work classifies fixed point data for circle actions on 6-dimensional oriented manifolds with isolated fixed points. It introduces a finite, constructive procedure that reduces the fixed point data to empty by performing equivariant connected sums at fixed points with $S^6$, $\mathbb{CP}^3$, and the 6-dimensional Hirzebruch analogues $Z_n$ (and their oppositely oriented counterparts). The approach hinges on a detailed analysis of weights and signs, together with relations among fixed points in the same isotropy submanifold to realize five operative moves. Termination is guaranteed by every step strictly lowering the largest weight, culminating in a fixed-point-free $S^1$-action. The results yield a combinatorial classification of fixed point data and have potential implications for similar analyses on almost complex or symplectic 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.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Example 1.3
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• proof