# Equivariant Principal Bundles over the 2-Sphere

## TL;DR

This paper classifies equivariant principal bundles over the 2-sphere with symmetry group actions, using isotropy representations and homotopy classes, and relates the classification to the first Chern class.

## Contribution

It introduces a classification framework for equivariant principal bundles over S^2 using isotropy representations and fixed point homotopy classes, linking to the first Chern class.

## Key findings

- Equivariant 1-skeletons are classified by isotropy representation data.
- Equivariant principal G-bundles are classified by fixed homotopy classes of maps.
- The underlying G-bundle is characterized by its first Chern class.

## Abstract

In this paper, we introduce the classification of equivariant principal bundles over the 2-sphere. Isotropy representations provide tools for understanding the classification of equivariant principal bundles. We consider a $\Gamma$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie group, and $\Gamma \subset SO(3)$ a finite group acting linearly on $S^2.$ We prove that the equivariant 1-skeleton $X \subset S^2$ over the singular set can be classified by means of representations of their isotropy representations. Then, we show that equivariant principal G-bundles over the $S^2 $ can be classified by a $\Gamma$-fixed set of homotopy classes of maps, and the underlying $G$-bundle $\xi$ over $S^2$ can be determined by first Chern class.