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Equivariant Principal Bundles over the 2-Sphere

Eyup Yalcinkaya

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 $Γ$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie group, and $Γ\subset SO(3)$ 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 $Γ$-fixed set of homotopy classes of maps, and the underlying $G$-bundle $ξ$ over $S^2$ can be determined by first Chern class.

Equivariant Principal Bundles over the 2-Sphere

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 -equivariant principal -bundle over with structural group a compact connected Lie group, and a finite group acting linearly on We prove that the equivariant 1-skeleton 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 can be classified by a -fixed set of homotopy classes of maps, and the underlying -bundle over can be determined by first Chern class.

Paper Structure

This paper contains 11 sections, 17 theorems, 43 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Split Equivariant Principal Bundles
  4. Equivariant Principal $G$-bundles
  5. 1-skeletons on $S^2$
  6. Classification of $G$-bundles Over 1-Skeletons On $S^2$
  7. Contractible Case
  8. Non-contractible Case
  9. Calculation of $Rep^G(\mathcal{I})$
  10. Equivariant Bundles on 1-skeletons
  11. $\Gamma$-$G$ Bundles over $S^2$

Key Result

Theorem 1.1

Let $\xi=(E,S^2,p,G,\Gamma)$ be a $\Gamma$-equivariant principal $G$-bundle over $S^2$ with a compact connected Abelian Lie group $G$ and $\Gamma\subset SO(3)$ be a finite subgroup acting on $S^2.$ A $\Gamma$-equivariant principal $G$-bundle over $S^2$ is determined by $Rep^G(\mathcal{I})$ and $c(\x

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Definition 2.2: Split $\Gamma$-Space
  • Definition 2.3: Isotropy Groupoid
  • Definition 2.4
  • Definition 2.5: Split Bundle
  • Definition 2.6: Isotropy Representation
  • Theorem 2.7
  • Proposition 2.8
  • ...and 21 more