Weights of circle actions on oriented manifolds with isolated fixed points

TL;DR

This paper proves that weights at fixed points of circle actions on oriented manifolds occur in pairs, removing the previous orientability assumption on isotropy submanifolds.

Contribution

It establishes the pairing of weights for circle actions on oriented manifolds without requiring isotropy submanifolds to be orientable.

Findings

Weights at fixed points occur in pairs for circle actions on oriented manifolds.

The pairing property holds without the orientability assumption on isotropy submanifolds.

The result extends previous claims to more general settings.

Abstract

For an action of the circle group $S^1$ on a compact oriented manifold with isolated fixed points, there is a claim that weights at the fixed points occur in pairs. This phenomenon holds for other types of $S^1$-manifolds, e.g., (almost) complex, symplectic, and unitary manifolds. A known proof of this claim assumes that the isotropy submanifolds are orientable. However, this assumption does not hold in general. In this note, we prove the claim without relying on that assumption.