\betaegin{equation}gin{document}

	
	\title{ Equivariant Principal Bundles over the 2-Sphere}
	
	
	\betaegin{equation}gin{abstract} 
		In this paper, we classify the equivariant principal bundles over the 2-sphere. 
		Isotropy representations provide tools for understanding the classification of equivariant principal bundles. We consider a $\Gammaamma$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie group, and $\Gammaamma \subset SO(3)$ 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 $\Gammaamma$-fixed set of homotopy classes of maps  and the underlying $G$-bundle $\xi$ over $S^2$ can be determined by first Chern class.
		
		\upsilonspace{.05in}
		
		\noindent {\em Keywords:} Equivariant principal bundles, Isotropy representation, Split bundles.
		
		\upsilonspace{.05in}
		
		\noindent {\em Mathematics Subject Classification 2010.} 
		Primary 55R91,55R15; Secondary 22A22.
		
	\end{abstract}
	
	\upsilonspace{.05in}
	\section{Introduction}

	
	
	Let $\Gammaamma$ and $G$ be Lie groups. A principal $(\Gammaamma, G)$-bundle $\xi$ over$B$ is a locally-trivial principal $G$-bundle $p: E \rightarrow B$ such that $E$ and $B$ are left $\Gammaamma$-spaces and the projection map
	is $\Gammaamma$-equivariant, so that  $\gammaammamma(e\cdot g) = (\gammaammamma e)\cdot g$ where $\gammaammamma \in \Gammaamma$ and $g \in G$ acting on $e \in E$.  We denote the bundle $\xi=(E,B,p,G,\Gammaamma).$ Equivariant principal bundles and their generalizations were studied by T. E. Stewart \cite {stewart} T. Tom Dieck \cite{tom}, R. Lashof \cite{lashof1}, P. May \cite {lashof2}, G. Segal \cite{lashof3}.  These authors use homotopy-theoretic methods. There exists a classifying space $B(\Gammaamma, G)$ for principal $(\Gammaamma, G)$-bundles \cite{tom}, so the classification of equivariant bundles
	in particular cases can be approached by studying the $\Gammaamma$-equivariant homotopy type of $B(\Gammaamma, G)$. If the structural group $G$ of the bundle is abelian, the main result of Lashof \cite{lashof1} states that equivariant bundles over a $\Gammaamma$-space B are classified by ordinary homotopy classes of maps $[X\timesmes_{\Gammaamma}E\Gammaamma, BG]$ as shown in \cite{lashof2}. Recently,  M. K. Kim \cite{2010arXiv1005.0681K} classifies  equivariant bundles over $2$-sphere by means of homotopy of  the set of equivariant clutching maps. Moreover, J. H. Verrette \cite{verrette2016results} studies  equivariant  vector  bundles  over the 2-sphere with effective actions by the rotational symmetries of the tetrahedron,  octahedron and  icosahedron for verifying Algebraic Realization Conjecture. 

	
	
	\indent Another approach is given by Hambleton and Hausmann \cite{ian1} for classifying equivariant principal bundles. They used the local invariants arising from isotropy representations at singular points of $(B,\Gammaamma)$, in the isotropy representation, at a $\Gammaamma$ fixed point $b_0 \in B$ it means the homomorphism $\alphapha_{b_0} : \Gammaamma \rightarrow G$ defined by the formula $$ \gammaammamma \cdot e_0 = e_0 \cdot \alphapha(\gammaammamma) $$
	where $e_0 \in p^{-1}(b_0).$ Denote the collection of isotropy representation of $\xi$ by $Rep^G_\Gammaamma(\mathcal{I}).$ The homomorphism $\alphapha$ is independent of the choice of $e_0$ up to conjugation in $G.$\\
	\indent In this article, $\Gammaamma$-equivariant principal $G$ bundles over $S^2$ will be classified by second approach. First of all, let $X \subset S^2$ be a $\Gammaamma$-invariant $1$-skeleton. The classification of $\Gammaamma$-equivariant principal $G$-bundle over $X$ will be determined by using the paper of Hambleton and Hausmann \cite{ian1} since $X$ is a split $\Gammaamma$-space. On the other hand, $S^2$ is not a split $\Gammaamma$-space. Therefore, we will use a different method to classify $\Gammaamma$-equivariant principal $G$-bundles over $S^2.$ Let $X \subset S^2$ be a $\Gammaamma$-equivariant $CW$- complex, then
	$$ X \xrightarrow{i} S^2 \xrightarrow{j} S^2 \cup CX \xrightarrow{k} \Sigmagma(X) \xrightarrow{m} \Sigmagma(S^2) \xrightarrow{\Sigmagma_i} \Sigmagma(S^2 \cup CX) \rightarrow \cdots $$ is a cofibration sequence where $CX$= cone on $X$ = $(X\timesmes[0,1])/{(a,0)\sigmam pt}$ and $\Sigmagma(S^2)=\text{suspension of} S^2.$
	
	\indent If we get homotopy classes of maps into space $B(\Gammaamma, G)$,then the following sequence
	$$ [\Sigmagma(S^2),Y] \xrightarrow{m^*} [\Sigmagma(X),Y] \xrightarrow{k^*} [S^2\cup CX, Y] \xrightarrow{j^*} [S^2,Y] \xrightarrow{i^*} [X,Y] $$ is a exact sequence of abelian groups since $B(\Gammaamma, G)$=$Y= \Omegaega Z$ is a loop space \cite{cost}. We will determine $[S^2,Y]$ by $\Sigmagma i^*$ and $i^*$ since $\Sigmagma i^*$ is the suspension map of $i^*.$ 
	
	
	\indent However, the classification of principal $G$-bundles over the
	2-sphere follows from the Steenrod equivalence theorem \cite{steen}. The only invariant is the "Chern number"  $c(\xi) \in [S^2,BG]= \pi_2(BG)=\pi_1(G).$ This paper is devoted to proving the following theorem.
	

	
	\betaegin{equation}gin{theorem} Let $G$ be a compact connected Lie group, $\Gammaamma\subset SO(3)$ be a finite subgroup acting linearly on $S^2.$ A $\Gammaamma$-equivariant principal $G$-bundle over $(S^2,\Gammaamma)$ is classified by $Rep_\Gammaamma^G(\mathcal{I})$ and $c(\xi) \in \pi_2(BG).$
	\end{theorem}
	\betaegin{equation}gin{corollary} If $Rep_\Gammaamma^G(\xi_1)\cong Rep_\Gammaamma^G(\xi_2)$ then $c(\xi_1)\equiv c(\xi_2)$  $mod\lvert \Gammaamma\lvert.$ 
	\end{corollary}
	\section{Preliminaries}

	\indent In this section, we give some definitions and
	theorems from the book of T.  Tom Dieck \cite{tom} which are used for classification at the next section. Now, we consider the group action as a left group action. If there exists a topological group acting on a
	topological space, we consider the isotropy group (stabilizer group) of each point in $X$ and the orbit space of the space $X$ under the group action.
	An action is called \emph{transitive} if for every $x_1,x_2$ $\in$ $X$ there exists a group element $g$ such that $gx_1$ = $x_2$ and the action is called \emph{free} if for every $x$ $\in$ $X$ the only element of $G$ (identity) fixes the element $x.$
	\betaegin{equation}gin{comment}
	\betaegin{equation}gin{df}[Isotropy Subgroup]
	Let $X$ be a topological space and $G$ be a group acting on $X$. For each $x \in X$, the \textbf{isotropy subgroup} of x is denoted by $G_x$ $=$ \{g $\in$ $G$ $\lvert$ g$\cdot$x$=$x\}.
	It is obvious that the $G_x$ is a subgroup of G.
	\end{df}
	\betaegin{equation}gin{df}[Orbit Space]
	Let $X$ be a topological space and $G$ be a group action on $X$. $Gx$:= \{gx $\lvert$ for all g $\in$ $G$\} is the \textbf{orbit} of $x$. The set of orbits induce an equivalence relation. The \textbf{orbit space} of $X$ is denoted by $G\betaackslash X$. Commonly, we  use $X/G$ for the orbit space if the action is defined before as a right action.
	\end{df}
	\indent For every subgroup $H$ of $G$ acting on $X,$ the fixed points set of $H$ is denoted by $X^H.$
	$$Fix(X,H) = X^H = \{x \in X \ \lvert h \cdot x = x \ \ \ \ \ for\ all\ h \in H,\ \ \}. $$
	
	\end{comment}
	A simplicial $G$-complex is \textbf{regular} if given elements $g_1, g_2, ... , g_n$ of the $G$ and two simplices $(v_1, v_2, ... , v_n)$ and $(g_1 \cdot v_1, g_2 \cdot v_2, ... , g_n \cdot v_n)$, there is an element $g\in G$ such that $gv_i={g_i}{v_i}$ for all i.
	Not all simplicial complexes are regular, but we can make a regular simplicial complex by using barycentric subdivision. For each $x\in X$, the \textbf{singular points set} of $X$ which is the isotropy subgroup of $x$ different than identity is denoted by $Sing(X,G) = \{ x \in X \lvert \ G_x \neq 1 \}.$
	\indent We say that an orbit which is equivalent to $G/H$ is of type $G/H.$ For $x$ $\in$ $S$ $\subset$ $M$, a manifold, such that $G_xS$ = $S$, $S$
	is a \textbf{slice} at x if the map
	$$\betaegin{equation}gin{array}[c]{cccc}
	G{\timesmes}_{G_x} S&\rightarrow&M\\
	(g,s)&\rightarrow&g\cdot s
	\end{array}$$
	is a a \textbf{tube} about $G$ $\cdot$ $x.$\\
	\indent Equivalently, we say that a slice at x is defined as a subspace $S$ of $X$ with the following
	properties:
	\betaegin{equation}gin{enumerate}[(i)]
		\item $S$ is closed in $G$ $\cdot$ $S$
		\item $G \cdot S$ is an open neighborhood of $G \cdot x$
		\item $G_x \cdot S = S$
		\item If $(g\cdot S)$ $\cap$ $S$ is non-empty, then $g$ is an element of $G_x$
	\end{enumerate}

	\indent 

	The following theorem says that there exists a slice under certain conditions. It was proven by Montgomery-Yang and Mostow [1957].
	\betaegin{equation}gin{theorem} \cite{bred} \lambdabel{thm: slice}
		Let $\Gammaamma$ be a compact Lie group and $X$ be a $\Gammaamma$-invariant space which is completely regular. Then there is a slice at each point and a tube around each orbit.
	\end{theorem}
	
	
	\indent After slicing theorem, the \emph{Riemann-Hurwitz Formula} is especially used in algebraic geometry. In this context, we will give this formula for group actions and use an analogous result for the graph.
	\betaegin{equation}gin{comment}
	
	\betaegin{equation}gin{theorem}[Riemann-Hurwitz Formula]
	Let $X$ be a compact connected oriented surface and $\Gammaamma$ $\timesmes$ $X$ $\rightarrow$ $X$ be an effective orientation preserving group action
	with the orbit space $X$/$\Gammaamma$ $\cong$
	$A$
	Then $$\chi(X)= |\Gammaamma| \chi(A)-\sum_{x\upsilonarepsilonsilon X}(|\Gammaamma_x|-1)$$
	\end{theorem}
	\end{comment}
	
	
	\betaegin{equation}gin{proposition}
		Let $X$ be a compact connected regular simplicial graph, $G$ be a finite group and $G$ $\timesmes$ $X$ $\rightarrow$ $X$ be a regular group action with the orbit space $X$/$G$ $\cong$
		$A$
		Then $$\chi(X)= |G| \chi(A)-\sum_{v_i\upsilonarepsilonsilon X}(|G_{v_i}|-1)$$
	\end{proposition}
	
	\betaegin{equation}gin{comment} 
	\betaegin{equation}gin{proof}
	Let the orbit space A be a graph with $v$ vertices and $e$ edges and $G$ be a group with the order $n$ and the singular set be $Sing(X,G)$ = $\{v_1,v_2,...,v_k\}.$ We relate the space $X$ with $v'$ vertices and $e'$ edges to the orbit space $A$.
	Since no edges are fixed by the group $G$, we can say that the number of edges of the space $X$ is $ne$. if the group $G$ acts freely on the $X$, then the number of vertices of $X$ would be $nv. $ On the other hand, we know that there exist some singular points with fewer pre-images. We should subtract them for the correct number of vertices of $X.$ For $v_i$ we counted it $|G_{v_{i}}|$ times. We should do this for all fixed points, then
	$$v'-e'= n(v-e)-\sum_{v_{i}\upsilonarepsilonsilon X}(|G_{v_{i}}|-1)$$
	$$\chi(X)= |G| \chi(A)-\sum_{v_i\upsilonarepsilonsilon X}(|G_{v_i}|-1)$$
	\end{proof}
	\end{comment}
	

	

	

	
	\subsection{Split Spaces}
	\indent We introduce the definition of a \emph{split $\Gammaamma$-space}
	since Hambleton and Hausmann
	\cite{ian1} basically show that if $X$ is \emph{split $\Gammaamma$-space} over $ A$,
	then there exists a bijection between the equivalence classes of split $\Gammaamma$-equivariant G-bundles over $X$ and $Rep^G(\mathcal I)$.
	\betaegin{equation}gin{definition}[Split $\Gammaamma$-Space] Let $\Gammaamma$ be a topological group and $A$ be a topological space. A $\Gammaamma$-space is a topological
		space equipped with a continuous left action of $\Gammaamma$. If the space $X$ is a $\Gammaamma$-space and $x \in X,$ we denote
		$\Gammaamma_{x}$ as the stabilizer of $x.$
		A \emph{ split $\Gammaamma$-space } over $A$ is denoted by a triple
		$(X, \pi, \upsilonarphi)$ where
		\betaegin{equation}gin{itemize}
			\item $X$ is a $\Gammaamma$-space.
			\item $\pi: X \rightarrow A$ is a continuous surjective map and, for each $a\in A$, the preimage
			$\pi^{-1}(a)$ is a single orbit.
			\item $\upsilonarphi$: A $\rightarrow$ X is a continuous section of $\pi$
		\end{itemize}
	\end{definition}
	\indent In this definition, we might prefer a \emph{ split $\Gammaamma$-space }  over $A$ by omitting the notation of maps $\pi$ and $\upsilonarphi$. Also, the map $\pi$ induces
	$\betaar{\pi}$: $\Gammaamma$ $\betaackslash$ $X$ $\rightarrow$ $A$ which is a homeomorphism since $\upsilonarphi$ provides its continuous inverse.
	\betaegin{equation}gin{comment} \subsection{Fiber Bundles}
	\indent Now, a fiber bundle will be introduced with Steenrod definitions \cite[p,6]{steen}. Actually, a manifold is locally isomorphic to $R^n$. The fiber bundle idea is the same as a manifold structure but the fiber bundle is locally isomorphic to cross product of open set and fiber. Let $E$ denote a bundle space, $B$ denote a base space, and $F$ denote a fiber. Let $ p: E \rightarrow B$ be a projection map if arbitrary open set $U$ is taken from E then the inverse image of $U$ under the projection map is locally homeomorphic to $U \timesmes F.$ We consider a space locally, that's why, in some context, the fiber bundle is called a \emph{locally trivial bundle}. Now, we shall give the definition of the fiber bundle more rigorously.
	\betaegin{equation}gin{df}[Fiber Bundle] A fiber bundle $\xi$ consists of a topological space $E$ called the bundle space, a topological space $B$ called base space,
	a continuous map $$p:E\rightarrow B$$ of $E$ onto $B$ called projection and a space $F$ called the fiber. The fiber over the point $x$ of $X$ is defined by $F_x$=$p^{-1}(x)$. Then it is necessary that each $F_x$ is homeomorphic to F. Therefore, for each $x\in X$,there is a neighborhood $U$ of $x$ and a
	homeomorphism $$\upsilonarphi: U\timesmes F\rightarrow p^{-1}(U)$$ such that $$p\upsilonarphi (x',y) = x' $$ where $x'$ $\in$ $X$, $y$ $\in$ $F$.
	\end{df}
	The fiber bundle can be illustrated as below, where the fiber is denoted by $F$, the bundle space is denoted by $E$, and the base space by $B.$
	$$\betaegin{equation}gin{array}[c]{ccc}
	F& \rightarrow&E\\
	&&\downarrow\scriptstyle{p}\\
	&&B
	\end{array}$$

	The first example is \textbf{product bundle} ,sometimes called trivial bundle, where the bundle space is $E$=$B$ $\timesmes$ $F$,\ \ \ $p(x,y)=x$, \ \ \ \ \ $x$ $\in$ $B$, $y$ $\in$ $F.$

	\indent The second example is Möbius line bundle $\mu$ defined to be a one-dimensional real vector bundle over the circle given as follows. Let $E=([0,1]\timesmes R)/{\sigmam}$ where $(0,t)\sigmam(1,-t)$ and
	$C$ be the middle circle $C=\{(s,0) \in E\}$. Then $\mu$ is the line bundle defined by the projection
	$$ p:E \rightarrow C \ \ \ \ \ where\ \ (s,t)\rightarrow (s,0)$$
	If we glue together the line $x=0$ and the line $x=1$ with the relation $ \sigmam$, we will get Möbius line bundle with the bundle space $E$. It can be seen the figure above the base space $C$ is the middle circle and the fiber of each point $(s,t)$ $\in$ $E$ is homeomorphic to the real line $R$.
	\betaegin{equation}gin{center}
	\includegraphics[height=40mm, width=60mm,]{mobious.png}
	\end{center}
	\betaegin{equation}gin{df}[Coordinate Bundle]  \cite[p,7]{steen}
	A coordinate bundle $\xi$ consists of a topological space $E$ called the bundle space, a topological space $B$ called base space,
	a continuous map $$p:E\rightarrow B$$ of $E$ onto $B$ called projection and a space $F$ called the fiber, an effective topological transformation group $G$ of $F$ called the group of the bundle, a family of $\{ U_j\}$ of open sets covering $B$ with an index set $J$, the $U_j$'s are called coordinate neighborhoods, and for each $j$ in $J$, a homomorphism
	$$\upsilonarphi_{j} :U_{j}\timesmes F\rightarrow p^{-1}(U_{j})$$
	called the coordinate function.
	\end{df}
	\indent The coordinate functions are required to satisfy the following conditions:
	$$p\upsilonarphi_j (x,y) = x \ \ \ \ \ \ \ \ \ \ for \ x \in U_j,\ y \in F$$
	\indent if the map $\upsilonarphi_{j,x}$ : $F \rightarrow p^{-1}(x)$ is defined by $\upsilonarphi_{j,x}(y)$ = $\upsilonarphi(x,y),$
	\newline then, for each pair $i,j$ in $J,$ and each $x$ $\in$ $U_{i} \cap U_{j},$ the homomorphism
	$$\upsilonarphi_{j,x}^{-1}\upsilonarphi_{i,x} : F \rightarrow F $$ overlaps with the operation of an element of $G$ and for each pair $i,j$ $\in$ $J$, the map
	$$ g_{ji} : U_i \cap U_j \rightarrow G $$ defined by $g_{ji}(x)$ = $ \upsilonarphi_{j,x}^{-1}\upsilonarphi_{i,x}$ is continuous. One can check that the functions $g_{ji}$'s satisfy the axioms of a group (closure, associative, an identity element, an inverse element, cocycle condition).

	\betaegin{equation}gin{df}[Vector Bundle]
	Let $\xi$ be a fiber bundle with the fiber $F$. The fiber bundle $\xi$ is called a \textbf{real vector bundle} if the fiber is $\mathbb{R}^n$ and the transformation group $G$ is the general linear group $GL_n(\mathbb{R}).$
	\end{df}
	\indent Here, if the group is $GL_n(\mathbb{C})$ and acting on $E$ and $F=\mathbb{C}$ then the fiber bundle is called a complex vector bundle.
	\end{comment}
	
	\betaegin{equation}gin{comment}
	
	\subsection{Equivariant Principal Bundle}
	\betaegin{equation}gin{df}[Principal Fiber Bundle]
	Let $\xi$ be a fiber bundle with the fiber $F$. The fiber bundle $\xi$ is called a \textbf{principal fiber bundle} if the fiber is homeomorphic to the transformation group $G$ and the transformation group $G$ acts freely on $F$ and $F=G.$
	\end{df}
	\betaegin{equation}gin{df}
	Let $\xi$ =($p$ : $E$ $\rightarrow$ $B)$ and $\xi'$ =( $p'$ : $E'$ $\rightarrow$ $B'$ ) be two principal $G$-bundles and let $u$ : $E$ $\rightarrow$ $E'$ and $f$ : $B$ $\rightarrow$ $B'$ be two maps such that $p'n=fp$.
	$$\betaegin{equation}gin{array}[c]{cccc}
	E& \xrightarrow{u}&E'\\
	\downarrow\scriptstyle{p} &&\downarrow\scriptstyle{p'}\\
	B&\xrightarrow{f}&B'
	\end{array}$$
	A pair ($u,f$) is called a principal bundle map (or just a principal map) if $u$ : $E$ $\rightarrow$ $E'$ is $G$-equivariant in the sense that
	$$ u(eg) = u(e)g $$ for all $e$ $\in$ $E$ and $g$ $\in$ $G.$
	\end{df}
	
	
	\betaegin{equation}gin{df}[Associated Bundle] Let $\xi$=($p$ : $E$ $\rightarrow$ $B$) be a fiber bundle and let $G$ be a left action on $F$ and let G be a right action on $E$. Then $\xi_p$ (=$E \timesmes_G F$ $\rightarrow$ $B$) is called an associated bundle of $\xi$ with fiber F.
	\end{df}
	\betaegin{equation}gin{df} Let $\xi$ and $\nu$ be two fiber bundles with the same base space $B$ and the same fiber $F.$ Then they are called equivalent if their associated bundles are equivalent.
	\end{df}
	\end{comment}
	
	
	
	
	
	\section{Classifying Equivariant Principal G-Bundle}

	\subsection{Split Equivariant Principal Bundles}
	\indent Now, we introduce the isotropy groupoid for given split $\Gammaamma$-space.
	Then we define the isotropy representation of an isotropy groupoid. Finally, we give the theorem which is proven by Hambleton and Hausmann
	in \cite{ian1}.

	\betaegin{equation}gin{definition}[Isotropy Groupoid] Let (X,$ \pi$,$\upsilonarphi$) be a split $\Gammaamma$-space over the space A. The \textbf{Isotropy groupoid} of $X$ is denoted by
		$$\mathfrak{I}(X):=\{(\lambdambda, a)\in \Gammaamma \timesmes A \lvert \ \lambdambda \in \Gammaamma_{\upsilonarphi(a)}\} $$
	\end{definition}
	\indent The isotropy groupoid of the space $X$ is the subspace of $\Gammaamma \timesmes A$ such that for each $a\in A$, the space
	$\mathfrak{I}_a= \mathfrak{I} \cap(\Gammaamma \timesmes \{a\})$ can be written as the form $\mathfrak{I}_a' \timesmes \{a\},$ where $\mathfrak{I}_a'$
	is a closed subgroup of $\Gammaamma.$\\
	\indent We denote by $A^{(n)}$ the skelata of $A,$ by $\Omegaega$ = $\Omegaega(A)$ the set of cells of A,  by $d(e)$ the dimension of a cell $e$ $\in$ $\Omegaega$ and $\Omegaega_n$ = $\{$ $e$ $\in$ $\Omegaega$ $\lvert$ $d(e)$ = $n$ $\}$  for the CW-complex $A.$ Also, we denote by $e(a)$ $\in$ $\Omegaega$ the cell $e$ of the smallest dimension such that $a$ $\in$ $A.$
	\betaegin{equation}gin{definition} A $(\Gammaamma, A)$ groupoid $\mathfrak{I}$ is called cellular if it is locally maximal and if $\widetildede{\mathfrak{I_a}} = \widetildede{\mathfrak{I_b}}$ when $e(a) = e(b).$
	\end{definition}
	\betaegin{equation}gin{definition}[Split Bundle] Let $(X, \pi,\upsilonarphi)$ be a split $\Gammaamma$-space over $A$ with isotropy groupoid $\mathfrak{I}$ and let $\xi = (p: E \rightarrow X)$ be a principal G-bundle. Then, the bundle $\xi$ is called a \textbf{split bundle} if $\upsilonarphi^*\xi$ is trivial.
	\end{definition}
	\indent A topological space is called a \emph{contractible space} if it is homotopic to a point. Moreover, it is known that every compact space is a paracompact space. All spaces which we use in this paper are compact spaces, then they are also paracompact.
	
	\indent Any $\Gammaamma$-equivariant principal G-bundle is a split bundle if the orbit space A is a contractible, paracompact space.
	This is used for classifying the equivariant principal bundles over $X$. For the finite subgroups of $SO(3)$, we separate conditions according to that  the space $A$ is contractible or not.
	Except for the orbit space of dihedral subgroup, all orbit spaces of subgroups of $SO(3)$ are contractible and paracompact. Hence, the theorem proven by Hambleton and
	Hausmann \cite{ian1} can be used for contractible case and we use a trick for the non-contractible case since the orbit space of non-contractible space is homeomorphic to $S^1.$

	\betaegin{equation}gin{definition}[Isotropy Representation] Let $G$ be a topological group and $\mathfrak{I}$ be a $(\Gammaamma,A)$ groupoid. A continuous representation of $\mathfrak{I}$ is continuous map
		$$\iota : \mathfrak{I}\rightarrow G$$ such that the restriction of $\iota$ to each point $a$ $\in$ $A$ is group homomorphism from $\mathfrak{I}_a$ to $G$. It can be denoted by
		$\iota_a$: $\mathfrak{I}_a$ $\rightarrow$ $G$.
	\end{definition}

	\indent A continuous representation of $\iota$ : $\mathfrak{I}$ $\rightarrow$ $G$ is called \emph{ locally maximal} if for each point $a$ $\in$ $A$, there exists a neighborhood $U$ such that $\mathfrak{I}_u$ is subgroup of $\mathfrak{I}_a$ for all $u$ $\in$ $U$.
	Moreover, the isotropy groupoid $\mathfrak{I}$ is called weakly locally maximal if there exists a continuous map $g: U \rightarrow G$ such that $\alphapha_u (\gammaammamma) = g(u) \alphapha_a (\gammaammamma) g(u)^{-1}$ for all $u\in U$ and $\gammaammamma \in \mathfrak{I}_u.$
	The set of conjugacy classes of locally maximal continuous representations of $\mathfrak{I}$ can be denoted $Rep^{G}(\mathfrak{I}).$ \\
	\indent Let $(X, \pi,\upsilonarphi)$ be a split $\Gammaamma$-space over the space A with isotropy groupoid $\mathfrak{I}.$ Let $\xi=(p: E \rightarrow X)$ be a split $\Gammaamma$-equivariant principal $G$-bundle over the space $X.$ There exists a continuous lifting $\widetildede{\upsilonarphi}^*(\xi) : A \rightarrow E$ of $\upsilonarphi$ since $\upsilonarphi^*(\xi)$ is trivial.  The equation
	$$
	\gammaammamma \widetildede{\upsilonarphi} (a) = \widetildede{\upsilonarphi}(a) \alphapha_a(\gammaammamma),$$
	(valid for $a \in A$ and $\gammaammamma \in \mathfrak{I_a})$ determines a continuous representation $\alphapha_{\xi,\widetildede{\upsilonarphi}} : \mathfrak{I} \rightarrow G.$
	Hambleton and Hausmann \cite{ian1} checked that $\alphapha_{\xi}$ does not depend on the choices $\widetildede{\upsilonarphi}$ and depends only on the $\Gammaamma$-equivariant isomorphism class of $\xi$. Then, the classification theorem is the following.

	\betaegin{equation}gin{theorem} [Classification Theorem of Hambleton and Hausmann] \cite{ian1} \lambdabel{class}
		Let $(X,\pi,\upsilonarphi)$ be a split $\Gammaamma$-space over the orbit space $A$ with the isotropy groupoid $\mathfrak {I}$. Assume that $A$ is locally compact, the group $\Gammaamma$ is a compact Lie group and $\mathfrak{I}$ is locally maximal. Then for any compact Lie group G, the map
		$$ \Phi: SBun_{\Gammaamma}^{G} \rightarrow Rep^{G}(\mathfrak{I})$$ is a bijection.
	\end{theorem}
	\indent The $S^2$ is the CW-complex but not a split $\Gammaamma$-space. For this reason, we investigate the restriction of $S^2.$ Let $X\subset S^2$ be a $\Gammaamma$-equivariant 1-$skeleton.$ The theorem \ref{class} can be used to classify equivariant principal $G$-bundles over $X$.  For the subgroups of the $SO(3)$ (except the dihedral group), the orbit spaces are
	contractible and paracompact. It follows that all equivariant bundles over $X$ are the split bundles. For the dihedral group we can use the same method. On the other hand, we cannot guarantee that all equivariant principal $G$ bundles are split bundles since the orbit space $A$ for the dihedral group is not contractible.
	In the theorem, the group is a compact Lie group. If the group G is abelian, then the non-split bundle space over the space $X$ is isomorphic to the bundle space over the orbit space $A$ since the projection maps are equivariant.
	\betaegin{equation}gin{proposition} \cite{ian1} Let $\Gammaamma$ be a compact Lie group and let 
		$(X,\pi,\upsilonarphi)$ be a split $\Gammaamma$-space over the orbit space $A$ with the isotropy groupoid $\mathfrak {I}.$
		Suppose that $\mathfrak {I}$ is locally maximal and that $A$ is a locally compact space. If the group $G$ is a compact abelian group, then there exists an isomorphism between abelian groups
		$$ (\Phi,\upsilonarphi*) : Bun^G_\Gammaamma(X) \rightarrow Rep^G(\mathfrak{I}) \timesmes Bun^G(A).$$
	\end{proposition}
	\subsection{1-skeletons of $S^2$} \lambdabel{section:a}
	
	\indent  After related definitions and theorems, we will introduce the first theorem of the classification of equivariant bundles over $2$-sphere. If the $\Gammaamma$-equivariant
	1-skeleton is chosen for finite subgroups of $SO(3)$, then it can be concluded that every $\Gammaamma$-\emph{invariant} 1-\emph{skeleton}  on $S^2$ is split $\Gammaamma$-space.
	Hence, if $X$ is a split $\Gammaamma$-space, then equivariant principal $G$-bundles over the CW-complex $X$ is classified by the theorem  \ref{class}.
	\betaegin{equation}gin{theorem}\lambdabel{splittheorem}
		Let $\Gammaamma \subset SO(3)$ be a subgroup acting on $S^2$ and $X$ be a  $\Gammaamma$-invariant 1-\emph{skeleton}
		on $S^2$. Then $X \subset S^2$ is a split $\Gammaamma$-space over $A$.
	\end{theorem}
	\betaegin{equation}gin{proof}
		\indent We prove the theorem by induction for cyclic  groups and the dihedral group. For the other subgroups of the $SO(3)$, the theorem is proved by direct computation. 

		\\
		\\
		\betaegin{equation}gin{comment} \emph{Subgroups of $SO(3)$}
		\betaegin{equation}gin{enumerate}[(i)]
		\item Cyclic Subgroups
		\item Dihedral Subgroups
		
		\item Tetrahedral Subgroup
		\item Octahedral Subgroup
		\item Icosahedral Subgroup
		\end{enumerate}

		\end{comment}
		\betaegin{equation}gin{enumerate}[(i)]
			
			
			\item Cyclic Subgroups

			\\ Let \emph{ $\mathfrak{C_n}$} be a \emph{1-skeleton } over $ S^2$ such that it is composed of $2$ vertices and $n$ edges , $ \mathbb{C}_n $ be a cyclic group containing n elements and
			acting on \emph{ $\mathfrak{C_n}$} and $E_n$ be the orbit space of \emph{ $\mathfrak{C_n}$} under the group action of $ \mathbb{C}_n $. We will look the CW- complex $\mathfrak{C_n}$ illustrated below as a projection from the north pole
			to the xy-plane.

			\indent The CW-complex \emph{ $\mathfrak{C_n}$} is a split $\mathbb C_n$-space. It is proven by induction. The orbit space $E_n$ and the group \emph{ $\mathfrak{C_n}$} satisfy the \emph{ proposition of Riemann-Hurwitz Formula}; 
			\betaegin{equation}gin{displaymath} \chi( \mathfrak{C_n})=n\chi(E_n)-\sum_{p\upsilonarepsilonsilon E_n}(|{\mathbb C_n}_{p}|-1)
			\end{displaymath}
			and the following holds by definition of Euler Characteristic;
			\betaegin{equation}gin{displaymath} \chi( \mathfrak{C_n})=2-n \  \chi(E_n)=1.
			\end{displaymath}
			The claim is that
			\betaegin{equation}gin{displaymath} \chi( \mathfrak {C_{n+1}})=\chi( \mathfrak{C_n})-1 \ \ \
			\ \ \ \ \chi(E_{n+1})=\chi(E_n).
			\end{displaymath}
			\indent After we add 1 edge to the \emph{ $\mathfrak{C_n}$}, the new CW-complex is the \emph{ $\mathfrak{C_{n+1}}$}. Furthermore, we will subtract 1 from Euler characteristic
			since vertices (the north  pole and the south pole) are fixed under new construction. On the other hand, the orbit space $E_n$ does not
			change since the orbit space $E_{n+1}$ is still composed of 1 edge and 2
			vertices.
			
			\indent if $n=2$, $\Gammaamma=\mathbb{C}_2$, then $\chi(\emph{ $\mathfrak{C_2}$})=0$, $\chi(E_2)=1$
			and \betaegin{equation}gin{displaymath} \chi( \zeta_n)=2-n
			\end{displaymath}
			\indent $\chi(\emph{ $\mathfrak{C_{n+1}}$})=1-n$ since the CW-complex \emph{ $\mathfrak{C_{n+1}}$} has $2$ vertices and
			$n+1$ edges. Therefore,
			\indent $\chi(\emph{ $\mathfrak{C_{n+1}}$})=\chi(\emph{ $\mathfrak{C_n}$})-1$, \ \ \emph{ $\mathfrak{C_{n+1}}$} and $E_{n+1}$ satisfies the
			\emph{proposition of Riemann-Hurwitz Formula}.
			
			\betaegin{equation}gin{figure}[H]
				\centering
				
				\includegraphics[height=12mm, width=40mm,]{1-skeleton.png}
				
				\caption{Orbit spaces of cyclic groups}
			\end{figure}
			It can be said that the isotropy groupoid of the CW-complex $\mathfrak{C_n}$ is $\mathfrak{I}$ cellular since it is defined by $\mathfrak{I}_0$=$\mathfrak{I}_1$=$ \mathbb{C}_n $ and $\mathfrak{I}_{01}$ $=$id. Then we can conclude that there is a continuous section from orbit space to $X.$

			\item Dihedral Subgroups
			\\ Let $\mathfrak D_n$ be a \emph{1-skeleton} over $ S^2$
			such that it is composed of $2n+2$ vertices, $6n$ edges since the CW-complex $\mathfrak D_n$ is actually composed of n-gon
			($n$ vertices and $n$ edges) stated at equator of a sphere.
			
			\indent Let $D_{2n}$ be a dihedral group of order $2n$ acting on the CW-complex
			$\mathfrak D_n$ and we should add $n+2$ vertices and
			$5n$ edges on the CW-complex $\mathfrak D_n$ to provide regular CW-complex where $n$ is the number of vertices of polygon.
			\indent Let $\mathcal D_n$ be the orbit space of $\mathfrak
			D_n$ under the group action of $D_{2n}$ .

			\indent The CW-complex $\mathfrak D_n$ is a split $D_{2n}$-space. It is proven by induction
			since the orbit space $\mathcal D_n$ and the CW-complex $\mathfrak D_n$ satisfy the
			\emph{ Riemann-Hurwitz Formula};
			\betaegin{equation}gin{displaymath} \chi(\mathfrak D_n)=2n\chi(\mathcal D_n)-\sum_{x\upsilonarepsilonsilon \mathfrak D_n}(|{D_{2n}}_p|-1)
			\end{displaymath} 
			and the following holds by definition of Euler Characteristic;
			\betaegin{equation}gin{displaymath} \chi(\mathfrak D_n)=(2n+2)-6n=2-4n \ \ \ \chi(\mathcal D_n)=0.
			\end{displaymath}
			The claim is that
			\betaegin{equation}gin{displaymath} \chi(\mathfrak D_{n+1})=\chi(\mathfrak D_n)-4 \  and \  \chi(\mathcal D_{n+1})=\chi(\mathcal D_n).
			\end{displaymath}
			\indent After we add one vertex to the CW-complex $\mathfrak D_n$, we will subtract four from the Euler characteristic of $\mathfrak D_n$ to construct $\mathfrak D_{n+1}$.
			Indeed, one vertex is not sufficient since it is necessary to construct
			a regular CW-complex. Therefore, it needs 2 extra vertices different than the first vertex which is
			added first. Otherwise it would not be a regular CW-complex.
			On the other hand, the orbit space does not
			change after adding vertices and edges since the orbit space is actually composed of $3$ edges and $3$
			vertices as well.
			\indent If $n=2$, \ \ \ then $\Gammaamma=D_4$,
			the CW-complex $\mathfrak D_2$ is $1$-skeleton on $S^2$ such that it has square stating on equator of
			$S^2$.
			\indent Let the vertices of the square be named 1, 2, 3, 4. \newline
			$$\Gammaamma=D_4= \{ (),(1234),(1423),(1432),(14)(23),(12)(34),(13),(24) \} $$
			then \betaegin{equation}gin{displaymath} \chi(\mathfrak D_2)=6 \ \ \ \ \ \ \chi(\mathcal D_2)=0 \end{displaymath}
			and \betaegin{equation}gin{displaymath} \chi(\mathfrak D_n)=2-4n
			\end{displaymath}
			\indent $\chi(\mathfrak D_{n+1})=-2-4n$ since $\mathfrak D_{n+1}$ has $ 2n+4$ vertices and
			$6n+6$ edges.
			Therefore,
			\betaegin{equation}gin{displaymath} \chi(\mathfrak D_{n+1})=\chi(\mathfrak D_n)-4,
			\end{displaymath} $\mathfrak D_{n+1}$ and $\mathcal D_{n+1}$ satisfy the
			\emph{Riemann-Hurwitz Formula}.
			
			\betaegin{equation}gin{figure}[H]
				\centering
				
				\includegraphics[height=25mm, width=65mm,]{triangle.png}
				\caption{Orbit space of the dihedral group}
			\end{figure}
			
			We can say that the isotropy groupoid of $\mathfrak{D_n}$ is $\mathfrak{I}$ since it is defined by $\mathfrak{I}_0$=$\mathfrak{I}_1$=$ \mathbb{C}_2 $, $\mathfrak{I}_2$=$ \mathbb{C}_n $and $\mathfrak{I}_{01}$ $=$ $\mathfrak{I}_{12}$=$\mathfrak{I}_{02}$=id. Then we can find a section from orbit space to base space.
			\item Tetrahedral Subgroup,
			Octahedral Subgroup,
			Icosahedral Subgroup 
		\end{enumerate}
		
		can be calculated by direct computations without induction. Detailed proof can be followed at \cite{yalcinkaya2012equivariant}.
		\betaegin{equation}gin{comment}

		\subsubsection{Tetrahedral Subgroup}
		\indent Let $\mathfrak T$ be a tetrahedron and $A_4$ be a tetrahedral group of order 12 acting on $\mathfrak T$ and let $T$ be
		an orbit space of $\mathfrak T$ under the group action of $A_4$.\\
		\\
		$A_4= \{ (), (12)(34),(13)(24),(14)(23),(123),(132),(124),(142),(134),(143), \\ (234),(243) \}$\\

		Firstly, a tetrahedron has 4 vertices, and vertices of $\mathfrak T$ are named by 1, 2, 3, 4.
		We will add extra vertices and edges to provide a regular CW-complex. It is necessary to put 6 vertices
		in the middle of edges and 4 vertices in the middle of the faces. Finally, $\mathfrak T$ has 14 vertices and 24 edges.
		$$\chi(\mathfrak T)=-10$$
		The tetrahedron has 4 vertex-rotation of order 3 (i.e. $(123),(142),(143),(243)$)
		and 3 edge-rotation order of 2 (i.e.$(12)(34),(13)(24),(14)(23)$).
		Therefore, the orbit space $T$ is composed of 2 vertices and 1 edge.
		The tetrahedron is split $A_4:$ space we can prove this by the \emph{Riemann-Hurwitz Formula}.

		$$-10=12\chi(T)-(4(2+2)+3(1+1)) $$
		$$\chi(T)=1$$
		\betaegin{equation}gin{center}
		\includegraphics[height=10mm, width=30mm,]{1-skeleton.png}
		\end{center}
		We can say that the isotropy groupoid of $\mathfrak{T}$ is $\mathfrak{I}$ cellular since it is defined by $\mathfrak{I}_0$=$ \mathbb{C}_2 $ $\mathfrak{I}_1$=$ \mathbb{C}_3 $ and $\mathfrak{I}_{01}$ $=$id. Then we can find a section from $orbit space$ to $X.$

		\subsubsection{Octahedral Subgroup}
		\indent Let $\mathcal C$ be a cube and $\mathfrak O$ be an octahedral group of order 24 and acting on the cube $\mathcal C$ and $O$ be the
		orbit space of $\mathcal C$ under the group action of $\mathfrak O$.
		The octahedral group $\mathfrak O$ can be considered as a subgroup of $S_8$.\\

		Firstly, the cube $\mathcal C$ has 8 vertices and 12 edges.
		We must add extra vertices and edges to satisfy the regularity of CW-complex.
		It is necessary to add 12 vertices in the middle of edges
		and 6 vertices in the middle of faces. Finally, the cube $\mathcal C$ has 26 vertices and 48 edges.
		$$\chi(\mathcal C)=-22$$
		The cube $\mathcal C$ has 4 vertex-rotation of order 3,
		6 edge-rotation of order 2,
		and 3 face-rotation of order 4.
		Therefore, the orbit space $O$ is composed of 3 vertices and 2 edges. The cube
		$\mathcal C$ is a split $\mathfrak O$-space since it satisfies the \emph{Riemann-Hurwitz Formula.}

		$$-22=24\chi(O)-(3(3+3)+4(2+2)+6(1+1))$$
		$$\chi(O)=1$$
		\betaegin{equation}gin{center}
		\includegraphics[height=20mm, width=60mm,]{2-1-skeleton.png}
		\end{center}
		We can say that the isotropy groupoid of the cube $\mathcal {C}$ is $\mathfrak{I}$ cellular since it is defined by $\mathfrak{I}_0$=$ \mathbb{C}_2 $, $\mathfrak{I}_1$=$ \mathbb{C}_3 $, $\mathfrak{I}_2$=$ \mathbb{C}_4 $, $\mathfrak{I}_{01}$ $=$id and $\mathfrak{I}_{12}$ $=$id. Then we can construct a section from $X.$

		\subsubsection{Icosahedral Subgroup}
		\indent Let $\mathcal I$ be a Icosahedral and $H$ be an icosahedral group of order 60 acting on $\mathcal I$ and let $I$
		be an orbit space of $\mathcal I$ under the group action of $H$. The icosahedral group
		$H$ can be considered as a subgroup of $S_{12}$.\\

		Firstly, the dodecahedron $\mathcal I$ has 12 vertices and 30 edges. (20 faces which is useful for adding extra vertices)
		We will add extra vertices and edges to provide regular CW-complex.
		It is necessary to put 30 vertices in the middle of edges
		and 20 vertices in the middle of faces. Finally, the dodecahedron $\mathcal I$ has 62 vertices and 120 edges.
		$$\chi( \mathcal I)=-58$$
		The dodecahedron $\mathcal I$ has 6 vertex-rotation of order 5,
		15 edge-rotation of order 2,
		and 10 face-rotation of order 3.
		Then, the orbit space I is composed of 3 vertices and 2 edges.
		Therefore, the dodecahedron $\mathcal I$ is a split $H$- space: we show this by the \emph{Riemann-Hurwitz Formula.}
		$$-22=60\chi(I)-(6(4+4)+15(2+2)+10(1+1))$$
		$$\chi(I)=1$$
		\betaegin{equation}gin{center}
		\includegraphics[height=20mm, width=60mm,]{2-1-skeleton.png}
		\end{center}
		It is said that the isotropy groupoid of the dodecahedron $\mathcal {I}$ is $\mathfrak{I}$ cellular since it is defined by $\mathfrak{I}_0$=$ \mathbb{C}_2 $, $\mathfrak{I}_1$=$ \mathbb{C}_3 $, $\mathfrak{I}_2$=$ \mathbb{C}_5 $, $\mathfrak{I}_{01}$ $=$id and $\mathfrak{I}_{12}$ $=$id. Then we can construct a section from orbit space to $X.$
		\end{comment}
		
		
	\end{proof}

	\indent Now, we know that for each finite subgroup of
	$SO(3),$ we can choose a \emph{1-skeleton} CW-complex on $S^2$ such
	that it splits over the orbit space $A$. Hambleton and Hausmann \cite{ian1}
	proved that there is a bijection between split bundle space of $X$
	and isotropy representation of the groupoid if the space $X$ splits
	under some restrictions.
	\subsection{Classification of G-bundle Over 1-Skeletons On $S^2$}
	\lambdabel{section:Classification of Principal G-bundle Over 1-Skeletons On $S^2$}
	\indent Although we know the general theory about the classification
	of equivariant principal bundle from \cite{ian1} and \cite{ian2}, the
	computation of $Rep^G(\mathcal I)$ is generally hard. On the
	other hand, we can compute $Rep^G(\mathcal I)$ since the isotropy
	groupoid of $X$ and the orbit spaces of $X$ under $\Gammaamma \subset SO(3)$ are simpler than the other spaces. In the proof of theorem \ref{splittheorem},
	we choose special 1-skeletons for each subgroup of $SO(3),$ such that each of 1-skeletons is $\Gammaamma$-space.
	After that, the orbit spaces are different than each other. The
	common point of the cyclic group, the tetrahedral group, the icosahedral group,
	the dodecahedral group is that their orbit spaces are homeomorphic to
	[-1,1]. Hence, their orbit spaces are contractible and paracompact.
	Every $\Gammaamma$-equivariant principal-G bundle is a \emph{split $\Gammaamma$-space} where the orbit space $A$ is a
	contractible and paracompact space. For the other case, the orbit space $A$ is a triangle
	(in other words A is a graph) then there is a bijection between split
	bundle space of the CW-complex $X$ and the isotropy representation of groupoid $\mathfrak{I}$. In
	this case, there exist some non-split bundle over the space $X$. If the group $G$ is abelian, then there is an
	isomorphism between the bundle spaces of $X$ and $Rep^G(\mathcal I)$
	$\timesmes$ $Bun^G(A).$\\
	\indent We conclude that all the orbit spaces are a paracompact space since all the orbit spaces are compact. The orbit spaces of the different CW-complexes are shown in the following figure. 
	\betaegin{equation}gin{figure}[H]
		\centering
		
		
		\includegraphics[height=38mm, width=150mm,]{complete.png}  
		
		
		
		\caption{Orbit spaces}
	\end{figure}
	
	
	\indent However, we shall separate two cases since one of the orbit spaces is not contractible:
	\betaegin{equation}gin{enumerate}[(i)]
		\item The orbit space A is contractible.
		\item The orbit space A is not contractible.
	\end{enumerate}
	\subsubsection{Contractible Case}
	\indent All of the equivariant bundles over the space $X$ are split bundles, since the orbit space of the space $X$ (where the group is one of the cyclic group, the tetrahedral group, the octahedral group, and icosahedral group) is contractible and paracompact.
	Therefore, we will use the classification theorem of Hambleton and Hausmann \cite{ian1}. Since, the possible specific spaces $(*$,$\pi$,$\upsilonarphi$ $)$ split over its orbit spaces. All the possible orbit spaces are locally compact, all the possible isotropy groupoid are locally maximal and the group $\Gammaamma$ is a compact Lie group. Thus
	$$ \Phi: SBun_{\Gammaamma}^{G} \rightarrow Rep^{G}(\mathfrak{I})$$ is a bijection for all case where the orbit space is contractible.
	\subsubsection{Non-contractible Case}
	The bijection given is available for the dihedral case since the CW-complex $\mathfrak D_n$ splits over the orbit space $\mathcal D_n$ as well. But we cannot guarantee that all equivariant principal $G$ bundles are split over the space $X$. (i.e. there may exist some non-split bundles over the space $X$). For the non-split bundles over the space $X$, we should restrict our conditions. If we choose an abelian group $G$, then we can use the proposition of the theorem.
	The space $(\mathfrak D_n,\pi,\upsilonarphi)$ is a split $D_{2n}$-space over $\mathcal D_n$ with the isotropy groupoid $\mathfrak{I}$, and $\mathcal D_n$ is locally compact and $\mathfrak{I}$ is locally maximal. Then for any abelian Lie group $G$ , one has an isomorphism of
	$$ (\Phi,\upsilonarphi*) : Bun^G_\Gammaamma(X) \rightarrow Rep^G(\mathfrak{I}) \timesmes Bun^G(A).$$
	\indent Since the orbit space $\mathcal D_n$ is a triangle, the orbit space $\mathcal D_n$ is homeomorphic to the $S^1$. We know the bundles over the circle $S^1$, it induce map
	$$ \overline{(\Phi,\upsilonarphi*)} : Bun^G_\Gammaamma(X) \xrightarrow{\rm \approx} Rep^G(\mathfrak{I}) \timesmes Bun^G(S^1).$$
	\indent If the group $G$ is a connected compact Lie group, it follows
	$$ [S^1,BG] \cong \pi_{1}(BG) \cong \pi_{0}(G) \cong 0 $$
	\indent If the group G is not connected but still a compact Lie group, it follows
	$$ [X,S^1] \cong \pi_{1}(BG) \cong \pi_{0}(G) .$$
	\indent Now, we will define some new concepts to can calculate $Rep^G(\mathfrak{I})$ easier.
	\section{Calculation of $Rep^G(\mathfrak{I})$}
	\indent Let $\iota$ : $\mathfrak{I}$ $\rightarrow$ $G$ be an isotropy representation and let $\mathfrak{I}$ be a $(\Gammaamma$,$A)$-groupoid. The isotropy representation $\iota$ is called cellular if $\iota_a = \iota_b$ when $e(a) = e(b).$ For each $e\in \Omegaega(A),$ this defines Hom($\mathfrak{I}_e$, $G)$ with face compatibility conditions $\iota_e$ = $\iota_f$ $\lvert$ $\mathfrak{I}_e$ whenever $f$ $\le$ $e.$ The set of conjugacy classes of cellular representations of $\mathfrak{I}$ into $G$ is denoted by $Rep^{G}_{cell}$ $(\mathfrak{I}).$
	\indent For a cellular representation $\iota$ : $\mathfrak{I}$ $\rightarrow$ $G$ and a cell $e$ of $A,$ we can associate its conjugacy class [$\iota_e$] $\in$ $\overline{Hom}$ ($\mathfrak{I}$,$G$). It follows that there is a map $\betaegin{equation}ta$
	$$ \betaegin{equation}ta : Rep_{cell}^G(\mathfrak{I}) \rightarrow \prod_ {e \in \Omegaega (A)} {\overline{Hom}(\mathfrak{I}_e, G).}$$
	If we pick an element $b_e$ from this product, it must satisfy the face compatibility condition. Now, we will define
	$$ \overline{ Rep}_{cell}^G(\mathfrak{I}) = \{(b_e) \in \prod_ {e \in \Omegaega (A)} \lvert \ \ \ \ b_e = b_f \lvert \mathfrak{I}_e \ \ if\  f \le e \}$$ and we can replace $\betaegin{equation}ta$ as a map $\betaegin{equation}ta$ : $Rep_{cell}^G(\mathfrak{I})$ $\rightarrow$ $\overline{Rep}_{cell}^G(\mathfrak{I}).$ Then the diagram is commutative,

	
	
	$$\betaegin{equation}gin{tikzcd}[column sep=small]
	Rep_{cell}^G(\mathfrak{I})\arrow{rr}{\tau} \arrow[swap]{dr}{\betaegin{equation}ta}& &Rep^G(\mathfrak{I}) \arrow{dl}{\upsilon}\\
	& \overline{Rep}_{cell}^G(\mathfrak{I}) & 
	\end{tikzcd}    $$
	when $\mathfrak{I}$ is proper $(\Gammaamma$,$A)$-groupoid.
	\betaegin{equation}gin{theorem} Let $X\subset S^2$ $A=X/\Gammaamma$
		Let $\Gammaamma$ $\subset$ $SO(3)$ and G be a topological group. Let $\mathfrak{I}$ be a $(\Gammaamma$,$A)$-groupoid, where A is an orbit space. Then $\betaegin{equation}ta$ : $Rep_{cell}^G(\mathfrak{I})$ $\rightarrow$ $\overline{Rep}_{cell}^G(\mathfrak{I})$
		is surjective.
	\end{theorem}
	\betaegin{equation}gin{proof} For finite subgroup $\Gammaamma\subset SO(3),$ the orbit spaces of subgroups are either a tree or not a tree (for the dihedral case the orbit space is triangle) . One can find the detailed proof in \cite{ian2} when the orbit space is tree. Now, we only prove the dihedral case. Let $b \in \overline{Rep}_{cell}^G(\mathfrak{I})$ and let $v$ be a vertex of $A.$ We choose $\iota_v \in Hom(\mathfrak{I_v},G)$ representing $b_v.$ For an edge $e$ between $v$ and $v'$ we define $\iota_e$ $\in$ $Hom(\mathfrak{I_e},G)$ by $\iota_e = \iota_v \lvert \  \mathfrak{I}(e).$ Since $b \in \overline{Rep}_{cell}^G(\mathfrak{I})$, we choose $\iota_v' \in Hom(\mathfrak{I_v'},G)$ where ${\iota_v'}_\lvert$ = $\iota_e.$ Therefore, we can construct a cellular representation $\iota^1$ over the tree $A(v,1)$ of the points of distance $\leq$ $1$ from $v.$ We construct $\iota^2$ over $A(v,2)$ by means of this method. Hence, we choose the points of distances $< 3$ from $v,$ when $A(v,3)$ is defined. Now, this defines $\iota \in Rep_{cell}^G(\mathfrak{I})$ with $\betaegin{equation}ta(\iota) = b.$
	\end{proof}
	\betaegin{equation}gin{proposition}
		Let $\mathfrak{I}$ is a $(\Gammaamma,A)$-groupoid, where $\Gammaamma$ $\subset$ $SO(3)$ and the orbit space $A$ is a graph. Let $G$ be a path-connected topological group. Then $\upsilon : Rep^G(\mathfrak{I})\rightarrow \overline{Rep}_{cell}^G(\mathfrak{I})$ is surjective.
	\end{proposition}
	\betaegin{equation}gin{proof}
		One can use the classification theorem and the paper of Hambleton and Hausmann \cite{ian1}.
	\end{proof}
	\betaegin{equation}gin{theorem}
		Let $\mathfrak{I}$ be a proper $(\Gammaamma,A)$-groupoid with  $\Gammaamma \subset SO(3)$ a finite topological group and let $A$ be an orbit space. Let $G$ be a compact connected Lie group. Then $\tau: Rep_{cell}^G(\mathfrak{I}) \rightarrow Rep^G(\mathfrak{I})$ is bijective. 
	\end{theorem}
	\betaegin{equation}gin{proof}
		Let $\tau(\alphapha)$ = $\tau(\alphapha')$ with the two cellular representations. One can see $\alphapha$ = $\alphapha'$ by taking the conjugate of the one of them. For detailed proof one can look \cite{ian2}. The map $\tau$ is surjective. If we choose $a \in Rep^G(\mathfrak{I})$ then in our case, it will be cellular representation since isotropy group is different than identity only vertices it is identity in edges.
	\end{proof}
	\indent Now, we investigate $\tau$ and $\betaegin{equation}ta$ but we have limited information about $\upsilon$. To do this, we induce from the paper of Hambleton and Hausmann \cite{ian2}. Let $\kappa_1, \kappa_2 : \mathfrak{I} \rightarrow G$ be two isotropy representations such that $\kappa_1 = \kappa_2 \lvert_{\mathfrak{I}_e} =: \kappa.$ Then $\tau^{-1} (\kappa_1,\kappa_2$) is in bijection with the set of double cosets $\pi_0(Z_{\kappa_1})\betaackslash\pi_0(Z_{\kappa})/\pi_0(Z_{\kappa_2}).$
	\section{Classification of Equivariant Bundles on $1$-skeleton}
	Now, we know the relation between $Bun^G_\Gammaamma(X)$ and $Rep^G(\mathfrak{I})$ and we can calculate $Rep^G(\mathfrak{I}).$ Hence, we classify the equivariant principal G-bundles over $X$ where $X$ $\subset$ $S^2$ is the 1-skeleton of a regular $\Gammaamma$ structure for $S^2.$ We shall consider $\Gammaamma = C_n, D_{2n}, A_4, \mathfrak O, H.$
	\betaegin{equation}gin{theorem} Let $X$ = \emph{ $\mathfrak{C_n}$} be a $\Gammaamma$-equivariant \emph{1-skeleton } over $ S^2$ with $2$ vertices and $n$ edges , $ C_n $ be a cyclic group of order n and
		acting on \emph{ $\mathfrak{C_n}$} and $E_n$ be the orbit space of \emph{ $\mathfrak{C_n}$} under the group action of $ C_n $ with isotropy groupoid $\mathfrak{I}_1.$ Then, there is a bijection
		$$ {Bun^G_{C_n}}(\mathfrak{C_n}) \rightarrow Rep^G(\mathfrak{I}_1) $$ and $$ Rep^G(\mathfrak{I}_1) \cong \overline{Rep}^G(\mathfrak{I}_1) \cong \overline{Hom}(C_n,G) \timesmes \overline{Hom}(C_n,G).$$
	\end{theorem}
	\betaegin{equation}gin{proof} \emph{ $\mathfrak{C_n}$} is a split $C_n$-space, all equivariant bundles are split bundles.
	\end{proof}
	\betaegin{equation}gin{theorem} Let $X = \mathfrak D_n$ be a \emph{1-skeleton} over $ S^2$
		with $2n+2$ vertices, $6n$ edges and let $D_{2n}$ be a dihedral group of order $2n$ acting on the CW-complex
		$\mathfrak D_n$ and let $\mathcal D_n$ be an orbit space with isotropy groupoid $\mathfrak{I}_5.$ If $G$ is connected, Then there is a bijection
		$$ Bun^G_{D_{2n}}(\mathfrak D_n) \rightarrow Rep^G(\mathfrak{I}_2) $$ and $$ Rep^G(\mathfrak{I}_2) \cong \overline{Rep}^G(\mathfrak{I}_2) \cong \overline{Hom}(C_2,G) \timesmes \overline{Hom}(C_2,G) \timesmes \overline{Hom}(C_n,G).$$
	\end{theorem}
	\betaegin{equation}gin{proof}
		\emph{$\mathfrak D_n$} is split $D_{2n}$-space and $$Bun^G_\Gammaamma(X) \rightarrow Rep^G(\mathfrak{I}) \timesmes Bun^G(A)$$ since the group $G$ is a connected compact Lie group, it follows
		$$ [S^1,X] \cong \pi_{1}(BG) \cong \pi_{0}(G) \cong 0.$$
	\end{proof}
	\betaegin{equation}gin{theorem} Let $X$ = $\mathfrak{T}$ be a tetrahedron and $A_4$ be the tetrahedral group of order 12 acting on $\mathfrak{T}$ and let $T$ be
		an orbit space of $\mathfrak T$ under the group action of $A_4$ with isotropy groupoid $\mathfrak{I}_3.$ Then, there is a bijection
		$$ Bun^G_{A_4}(\mathfrak{T}) \rightarrow Rep^G(\mathfrak{I}_3) $$ and $$ Rep^G(\mathfrak{I}_3) \cong \overline{Rep}^G(\mathfrak{I}_3) \cong \overline{Hom}(C_2,G) \timesmes \overline{Hom}(C_3,G).$$
	\end{theorem}
	\betaegin{equation}gin{proof}
		$\mathfrak{T}$ is a split $A_4$-space, all equivariant bundles are split bundle.
	\end{proof}
	\betaegin{equation}gin{theorem} Let $X$ = $\mathcal C$ be a cube and $\mathfrak O$ be the octahedral group of order 24 and acting on the cube $\mathcal C$ and $O$ be the orbit space of $\mathcal C$ under the group action of $\mathfrak O$ with isotropy groupoid $\mathfrak{I}_4.$ Then, there is a bijection
		$$ Bun^G_{\mathfrak O}(\mathcal C) \rightarrow Rep^G(\mathfrak{I}_4) $$ and $$ Rep^G(\mathfrak{I}_4) \cong \overline{Rep}^G(\mathfrak{I}_4) \cong \overline{Hom}(C_2,G) \timesmes \overline{Hom}(C_3,G) \timesmes \overline{Hom}(C_4,G).$$
	\end{theorem}
	\betaegin{equation}gin{proof}
		\emph{ $\mathcal{C}$} is a split $\mathfrak O$-space, all equivariant bundles are split bundle.
	\end{proof}
	\betaegin{equation}gin{theorem} Let $X$ = $\mathcal I$ be an icosahedron and $H$ be an icosahedral group of order 60 acting on $\mathcal I$ and let $I$
		be an orbit space of $\mathcal I$ under the group action of $H$ with isotropy groupoid $\mathfrak{I}_5.$ Then, there is a bijection
		$$ Bun^G_H (\mathcal I) \rightarrow Rep^G(\mathfrak{I}_5) $$ and $$ Rep^G(\mathfrak{I}_5) \cong \overline{Rep}^G(\mathfrak{I}_5) \cong \overline{Hom}(C_3,G) \timesmes \overline{Hom}(C_4,G) \timesmes \overline{Hom}(C_5,G).$$
	\end{theorem}
	\betaegin{equation}gin{proof}
		\emph{ $\mathcal{I}$} is a split $H$-space, all equivariant bundles are split bundle.
	\end{proof}
	\section{Classification of $\Gammaamma$-$G$ Bundles over $S^2$}
	Let $X \subset S^2$ be a $\Gammaamma$-equivariant $1$-skeleton, then $\Gammaamma$-equivariant principal $G$-bundles over $X$ are classified by the isotropy representation since $X$ is split $\Gammaamma$-space. However, for $S^2$, we use a different technique to determine $\Gammaamma$-equivariant principal $G$-bundles over $S^2$ since $S^2$ is not a split $\Gammaamma$-space. Now, let $X \subset S^2$ be a $\Gammaamma$-equivariant $1$-skeleton, then
	\betaegin{equation}gin{equation}  X \xrightarrow{i} S^2 \xrightarrow{j} S^2 \cup CX \xrightarrow{k} \Sigmagma(X) \xrightarrow{m} \Sigmagma(S^2) \xrightarrow{l} \Sigmagma(S^2 \cup CX) \rightarrow \cdots \end{equation} is the cofibration sequence of $\Gammaamma$-equivariant $CW$-complexes where $CX$= cone on $X = (X\timesmes[0,1])/{(a,0)\sigmam pt}$ and $\Sigmagma(S^2)$= suspension of $S^2 = (X\timesmes[-1,1])/{(a,-1)\sigmam pt,\ (a,1)\sigmam pt}$ 
	
	\indent The $\Gammaamma$-fixed set of homotopy classes of maps into space $B(\Gammaamma,G)$,then the following sequence
	\betaegin{equation}gin{equation} \lambdabel{eq:solve} [\Sigmagma(S^2),Y] \xrightarrow{m^*} [\Sigmagma(X),Y] \xrightarrow{k^*} [S^2\cup CX,Y] \xrightarrow{j^*} [S^2,Y] \xrightarrow{i^*} [X,Y] \end{equation}is the exact sequnce of abelian groups provided that $ B(\Gammaamma,G)= Y= \Omegaega Z$ is a loop space which is defined by Costenable and Waner at \cite{cost}. Now, $[S^2,Y]$ is determined by $l^*$ and $i^*.$\\
	\indent Here, $S^2\cup CX \sigmameq \betaigvee S^2$ (induced from 2-cells) and 
	$\Sigmagma(X) \sigmameq \betaigvee S^2$ (induced from 1-cells) 
	then the exact sequence at \ref{eq:solve} will be the following   
	$$ [\betaigvee S^2,Y] \xrightarrow{k^*} [\betaigvee_{2-cells} S^2,Y] \xrightarrow{j^*} [S^2,Y] .$$
	\betaegin{equation}gin{equation}  \lambdabel {eq:solve1}[\betaigvee S^2,Y]_{\Gammaamma \neq D_{2n}}  \sigmameq \betaigoplus_{1-\chi(X)}\pi_2(BG)\ and\ \Gammaamma\ acts\ on\ product\ = I\otimes\pi_1(G)\end{equation} as a $\Gammaamma$-module
	$$ [\betaigvee S^2,Y] \xrightarrow{k^*} [\betaigvee_{2-cells} S^2,Y] \xrightarrow{j^*} [S^2,Y] $$  
	\betaegin{equation}gin{equation}  \lambdabel {eq:solve1}[\betaigvee S^2,Y]_{\Gammaamma = D_{2n}}  \sigmameq \betaigoplus_{1-\chi(X)}\pi_2(BG)\ and\ \Gammaamma\ acts\ on\ product\ = (I\oplus \mathbb{Z}\Gammaamma)\otimes\pi_1(G)\ \end{equation} as a $\Gammaamma$-module
	
	\betaegin{equation}gin{equation} \lambdabel {eq:solve2}[\betaigvee_{2-cells} S^2,Y]_\Gammaamma \sigmameq \betaigoplus_\mathcal{N} \pi_2(BG)\ and\ \Gammaamma\ acts\ on\ product\ = \mathbb{Z}\Gammaamma\otimes\pi_1(G)\ \end{equation} as a $\Gammaamma$-module provided that the ideal $I =\mathbb{Z}\{(\gammaammamma-1)\lvert \ \gammaammamma \in \Gammaamma \} \subset \mathbb{Z}\Gammaamma.$ 
	
	\indent For the equation  (\ref {eq:solve1}), the number of copy of $\pi_2(BG)$  is counted by rotations and the order of groups. For cyclic groups, the south pole and the north pole are fixed and there are $n$ edges such that vertices of edges are the south pole and the north pole. Then if 1 edge is collapsed to the point, then other edges turn to circles and we obtain (n-1) circles. For the dihedral group, since there is 1 orbit with 2n elements after 1 edge is collapsed to the point, then we obtain (2n-1) circles. The other 2n orbits have 2 elements, after 1 edge is collapsed to the point. Therefore, we obtain (4n-1) circles. For the tetrahedral group, there are 4 vertex rotations with the order 3, after 1 edge is collapsed to the point for each rotation, we obtain 8 circles. There are 3 edge rotations with the order 2. We obtain 3 circles. In total, we have 11 circles.  For octahedral group, there are 4 vertex rotations with the order 3, after 1 edge is collapsed to the point for each rotation, we obtain 8 circles. There are 6 edge rotations with the order 2. We obtain  6 circles. There are 3 face rotations with the order 4. After collapsing, we obtain 9 circles. In total, we obtain 23 circles. Therefore, we use same idea for the icosahedral group, we have 59 circles. Briefly, we say that the number of circles is $1-\chi( X)$  where $X\subset S^2$ is $\Gammaamma$-equivariant 1-skeleton. For the equation (\ref{eq:solve2}),
	the number of copy of $\pi_2(BG)$  is denoted by $\mathcal{N}$ and depends on the number of orbits and the order of group $\Gammaamma$. For the dihedral group, there are two orbits. For other finite subgroups of $SO(3)$ have a single orbit.
	
	\upsilonspace{.05in}
	
	
	\betaegin{equation}gin{table}[ht]
		\centering
		\betaegin{equation}gin{tabular}{l*2|c|r}
			Group & 1-$\chi(X)$ & $\mathcal{N}$ \\
			\hline
			Cyclic group & $n-1$ & $n$ \\
			Dihedral group & $4n-1$ & $4n$ \\
			Tetrahedral group & 11 & 12 \\
			Dodecahedral group & 23 & 24 \\
			Icosahedral group & 59 & 60 \\
			
			
		\end{tabular}
		\caption{The number of copy of $\pi_2(BG)$}
		
	\end{table}
	
	\upsilonspace{.3in}
	
	Since $k^*$ is an injective map, the following map holds; 
	$$0 \rightarrow I\otimes\pi_1(G) \xrightarrow{k^*} \mathbb{Z}\Gammaamma\otimes\pi_1{G} \xrightarrow{j^*} \mathbb{Z}\otimes\pi_1(G) $$ or $$0 \rightarrow (I\oplus\mathbb{Z}\Gammaamma)\otimes\pi_1(G) \xrightarrow{k^*} (\mathbb{Z}\Gammaamma\oplus\mathbb{Z}\Gammaamma)\otimes\pi_1{G} \xrightarrow{j^*} \mathbb{Z}\otimes\pi_1(G) $$and
	$Coker(k^*)$ $\sigmameq$ $\mathbb{Z}\otimes\pi_1(G)$ $\sigmameq$ $[S^2,B(\Gammaamma,G)]_\Gammaamma.$
	
	$$[\Sigmagma(X),Y] \xrightarrow{k^*} [S^2\cup CX,Y] \xrightarrow{j^*} [S^2,Y] \xrightarrow{i^*} [X,Y] $$

	\indent Now, let $Z$ and $Y$ be two $\Gammaamma$-space and $f:Z\rightarrow Y$ be continuous. Define $f^\gammaammamma(z)=\gammaammamma^{-1}f(\gammaammamma z)$ $f\rightarrow f^{\gammaammamma}$ gives an action of $\Gammaamma$ on $[Z,Y]$. Then $f=f^{\gammaammamma}$ $\leftrightarrow$ $\gammaammamma f(z)=f(\gammaammamma z)$ $\leftrightarrow$ $f$ is a $\Gammaamma$-map. Therefore, we shall say the following $[Z,Y]_\Gammaamma= Fix(\Gammaamma,[Z,Y])$ .
	
	\indent $Fix(\Gammaamma, I)=\{x \in I \lvert \gammaammamma x = x\} =0$  and we will determine $Fix(\Gammaamma, \mathbb{Z}\Gammaamma ).$   Let  $  t \in \Gammaamma $ be a generator. $\Gammaamma$ is acting on  $\mathbb{Z}\Gammaamma = \mathbb{Z} \oplus \mathbb{Z}t \oplus \cdots \oplus \mathbb{Z}t^{n-1}$  where $\lvert \Gammaamma \lvert =n$. The fixed set of $\mathbb{Z}\Gammaamma$ will be determined by $(a_0+a_1t+\cdots +a_{n-1}t^{n-1} )\gammaammamma =(a_0+a_1t+\cdots +a_{n-1}t^{n-1} )$ $\mathfrak orall \gammaammamma \in \Gammaamma$ implies that fixed elements are $\mathbb{Z}(1+t+\cdots+ t^{n-1})$. Then,  $Fix(\Gammaamma, \mathbb{Z}\Gammaamma )= \mathbb{Z}(1\sum_{\gammaammamma \in \Gammaamma}\gammaammamma).$ Therefore, 
	\betaegin{equation}gin{equation} \lambdabel {eq:solve6}[\Sigmagma X, Y]_\Gammaamma = Fix(\Gammaamma,I\otimes\pi_2Y)=0 \end{equation} and 
	\betaegin{equation}gin{equation} \lambdabel {eq:solve7}[S^2\cup C(X),Y]_\Gammaamma =Fix(\Gammaamma,\mathbb{Z}\Gammaamma\otimes \pi_2Y)= \pi_2(Y), \end{equation} since  we have $[S^2,Y]=\pi_2(Y)$ in the sequence (\ref{eq:solve}).   
	\betaegin{equation}gin{theorem} Let $G$ be a compact Lie group, $\Gammaamma \subset SO(3)$ finite and $\xi$ be a principal G-bundle over $S^2$.  $\Gammaamma$-equivariant principal $G$-bundle over $(S^2,\Gammaamma)$ is classified by $Rep_\Gammaamma^G(\xi)$ and $c(\xi) \in \pi_2(BG).$
	\end{theorem}
	\betaegin{equation}gin{proof} Let $[\nu]$ and $[\xi]$ $\in$ $[S^2,Y]_\Gammaamma .$  $[S^2,Y]_\Gammaamma \xrightarrow{i^*} [X,Y]_\Gammaamma $ and $[X,Y]_\Gammaamma \cong Rep^G_\Gammaamma (\mathfrak{I})$ If $Rep^G_\Gammaamma (\nu ) \ncong Rep^G_\Gammaamma (\xi)$ then one concludes that they are non-equivariant to each other. if $Rep^G_\Gammaamma(\nu)\cong Rep^G_\Gammaamma(\xi)$ then  $$ [\Sigmagma X, Y]_\Gammaamma \rightarrow [S^2\cup C(X),Y]_\Gammaamma \rightarrow [S^2,Y]_\Gammaamma$$ and by (\ref{eq:solve6}) and (\ref{eq:solve7}) $$ 0 \rightarrow \pi_2(G) \xrightarrow{\lvert \Gammaamma \lvert} [S^2,Y]_\Gammaamma $$ then we have they will be determined by first Chern class.
	\end{proof}
	\betaegin{equation}gin{corollary} If $Rep_\Gammaamma^G(\xi_1)\cong Rep_\Gammaamma^G(\xi_2)$ then $c(\xi_1)\equiv c(\xi_2)$   $mod\  \lvert \Gammaamma\lvert.$ 
	\end{corollary}
	
	\indent This theorem completes the classification of
	equivariant principal bundles over the 2-sphere.  Later studies will be focusing on how we can generalize this theorem to the $n$-sphere by these ideas.
	
	\upsilonspace{.05in}
	
	\textbf{Acknowledgements.} I would like to thank Prof. Dr. Ian Hambleton as my supervisor to teach tools of this article and  I would like to thank M. Kalafat for 
	motivating me to publish on this article and E. Yalcin for useful discussions.  This work is partially supported by the grant of McMaster University.
	\betaigskip
	
	\betaibliographystyle{amsplain}
	\betaibliography{Equivariant_Principal_Bundles_over_the_2-Sphere}
	
	\betaigskip
	
	{\small 
		\betaegin{equation}gin{flushleft} \textsc{Orta Do\u gu Tekn\' \i k  \" Un\' \i vers\' ites\' i, 06800, Ankara, T\" urk\' iye.}\\
			\textit{E-mail address:}  \texttt{\textbf{e142649@\,metu.edu.tr}}

		\end{flushleft}
	}
	

\end{document}