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`http://hdl.handle.net/11375/12063`

Title: | Equivariant Principal Bundles over the 2-Sphere |

Authors: | YALCINKAYA, EYUP |

Advisor: | Hambleton, Ian |

Department: | Mathematics and Statistics |

Keywords: | Equivariant principal bundles over $2$ sphere;Geometry and Topology;Mathematics;Geometry and Topology |

Publication Date: | 2012 |

Abstract: | <p>Isotropy representations provide powerful tools for understanding the classification of equivariant principal bundles over the $2$-sphere. 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 $S^2.$ Let $X$ be a topological space and $\Gamma$ be a group acting on $X.$ An isotropy subgroup is defined by $\Gamma_x = \{\gamma \in \Gamma \lvert \gamma x=x\}.$ Assume $X$ is a $\Gamma$-space and $A$ is the orbit space of $X$. Let $\varphi: A\rightarrow X$ be a continuous map with $\pi \circ \varphi = 1_A$. An isotropy groupoid is defined by $\mathfrak{I} = \{(\gamma,a) \in \Gamma\times A \lvert \ \gamma \in \Gamma_{\varphi(a)}\}.$ An isotropy representation of $\mathfrak{I}$ is a continuous map $\iota : \mathfrak{I} \rightarrow G$ such that the restriction map $\mathfrak{I}_a \rightarrow G$ is a group homomorphism. $\Gamma$- equivariant principal $G$-bundles are studied in two steps; \begin{enumerate} [1)] \item the restriction of an equivariant bundle to the $\Gamma$ equivariant 1-skeleton $X \subset S^2$ where $\mathfrak{I}$ is isotropy representation of $X$ over singular set of the $\Gamma$-sets in $S^2$ \item the underlying $G$-bundle $\xi$ over $S^2$ determined by $c(\xi)\in \pi_2(BG).$ \end{enumerate}</p> |

URI: | http://hdl.handle.net/11375/12063 |

Identifier: | opendissertations/6980 8027 2848337 |

Appears in Collections: | Open Access Dissertations and Theses |

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