<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> / Master of Science (MSc)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/12063 |
| Date | January 2012 |
| Creators | YALCINKAYA, EYUP |
| Contributors | Hambleton, Ian, Mathematics and Statistics |
| Source Sets | McMaster University |
| Detected Language | English |
| Type | thesis |
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