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.

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.

Tables

Table 1. Table 1: The number of copy of π 2 ​ ( B ​ G ) subscript 𝜋 2 𝐵 𝐺 \pi_{2}(BG)

Group	1-$\chi \left(X\right)$	$𝒩$
Cyclic group	$n-1$	$n$
Dihedral group	$4n-1$	$4n$
Tetrahedral group	11	12
Dodecahedral group	23	24
Icosahedral group	59	60

Equations

$$\gamma \cdot e_0=e_0\cdot \alpha (\gamma )$$

$$X\stackrel{i}{\to }{S}^2\stackrel{j}{\to }{S}^2\cup CX\stackrel{k}{\to }\Sigma (X)\stackrel{m}{\to }\Sigma (S^2)\stackrel{{\Sigma }_i}{\to }\Sigma (S^2\cup CX)\to \cdots$$

$$[\Sigma (S^2),Y]\stackrel{m^{\ast }}{\to }[\Sigma (X),Y]\stackrel{k^{\ast }}{\to }[S^2\cup CX,Y]\stackrel{j^{\ast }}{\to }[S^2,Y]\stackrel{i^{\ast }}{\to }[X,Y]$$

$$\begin{array}[c]{cccc}G{\times}_{G_{x}}S&\rightarrow&M\\ (g,s)&\rightarrow&g\cdot s\end{array}$$

$$\chi (X)=|G|\chi (A)-\sum _{v_iϵX}(|G_{v_i}|-1)$$

$$\mathfrak{I}(X):=\{(\lambda ,a)\in \Gamma \times A|\lambda \in {\Gamma }_{\phi (a)}\}$$

$$\iota :\mathfrak{I}\to G$$

$$\gamma \tilde{\phi }(a)=\tilde{\phi }(a){\alpha }_a(\gamma ),$$

$$\Phi :SBu{n}_{\Gamma }^G\to Re{p}^G(\mathfrak{I})$$

$$(\Phi ,\phi \ast ):Bu{n}_{\Gamma }^G(X)\to Re{p}^G(\mathfrak{I})\times Bu{n}^G(A).$$

$$\chi ({\mathfrak{C}}_{\mathfrak{n}})=n\chi (E_n)-\sum _{pϵE_n}(|{{\mathbb{C}}_n}_p|-1)$$

$$\chi ({\mathfrak{C}}_{\mathfrak{n}})=2-n\chi (E_n)=1.$$

$$\chi ({\mathfrak{C}}_{\mathfrak{n}+1})=\chi ({\mathfrak{C}}_{\mathfrak{n}})-1\chi (E_{n+1})=\chi (E_n).$$

$$\chi ({\zeta }_n)=2-n$$

$$\chi ({\mathfrak{D}}_n)=2n\chi ({\mathcal{D}}_n)-\sum _{xϵ{\mathfrak{D}}_n}(|{D_{2n}}_p|-1)$$

$$\chi ({\mathfrak{D}}_n)=(2n+2)-6n=2-4n\chi ({\mathcal{D}}_n)=0.$$

$$\chi ({\mathfrak{D}}_{n+1})=\chi ({\mathfrak{D}}_n)-4and\chi ({\mathcal{D}}_{n+1})=\chi ({\mathcal{D}}_n).$$

$$\Gamma =D_4=\{(),(1234),(1423),(1432),(14)(23),(12)(34),(13),(24)\}$$

$$\chi ({\mathfrak{D}}_2)=6\chi ({\mathcal{D}}_2)=0$$

$$\chi ({\mathfrak{D}}_n)=2-4n$$

$$\chi ({\mathfrak{D}}_{n+1})=\chi ({\mathfrak{D}}_n)-4,$$

$$\Phi :SBu{n}_{\Gamma }^G\to Re{p}^G(\mathfrak{I})$$

$$(\Phi ,\phi \ast ):Bu{n}_{\Gamma }^G(X)\to Re{p}^G(\mathfrak{I})\times Bu{n}^G(A).$$

$$\overline{(\Phi ,\phi \ast )}:Bu{n}_{\Gamma }^G(X)\stackrel{\approx }{\to }Re{p}^G(\mathfrak{I})\times Bu{n}^G(S^1).$$

$$[S^1,BG]\cong {\pi }_1(BG)\cong {\pi }_0(G)\cong 0$$

$$[X,S^1]\cong {\pi }_1(BG)\cong {\pi }_0(G).$$

$$\beta :Re{p}_{cell}^G(\mathfrak{I})\to \prod _{e\in \Omega (A)}\overline{Hom}({\mathfrak{I}}_e,G).$$

$${\overline{Rep}}_{cell}^G(\mathfrak{I})=\{(b_e)\in \prod _{e\in \Omega (A)}|b_e=b_f|{\mathfrak{I}}_{e}iff\le e\}$$

$$Re{p}_{cell}^G(\mathfrak{I})$$

$$Bu{n}_{C_n}^G({\mathfrak{C}}_{\mathfrak{n}})\to Re{p}^G({\mathfrak{I}}_1)$$

Full text

Keywords: Equivariant principal bundles, Isotropy representation, Split bundles.

Mathematics Subject Classification 2010. Primary 55R91,55R15; Secondary 22A22.

1 Introduction

Let $\Gamma$ and $G$ be Lie groups. A principal $(\Gamma,G)$-bundle $\xi$ over$B$ is a locally-trivial principal $G$-bundle $p:E\rightarrow B$ such that $E$ and $B$ are left $\Gamma$-spaces and the projection map is $\Gamma$-equivariant, so that $\gamma(e\cdot g)=(\gamma e)\cdot g$ where $\gamma\in\Gamma$ and $g\in G$ acting on $e\in E$. We denote the bundle $\xi=(E,B,p,G,\Gamma).$ Equivariant principal bundles and their generalizations were studied by T. E. Stewart [10] T. Tom Dieck [11], R. Lashof [6], P. May [7], G. Segal [8]. These authors use homotopy-theoretic methods. There exists a classifying space $B(\Gamma,G)$ for principal $(\Gamma,G)$-bundles [11], so the classification of equivariant bundles in particular cases can be approached by studying the $\Gamma$-equivariant homotopy type of $B(\Gamma,G)$. If the structural group $G$ of the bundle is abelian, the main result of Lashof [6] states that equivariant bundles over a $\Gamma$-space B are classified by ordinary homotopy classes of maps $[X\times_{\Gamma}E\Gamma,BG]$ as shown in [7]. Recently, M. K. Kim [5] classifies equivariant bundles over $2$-sphere by means of homotopy of the set of equivariant clutching maps. Moreover, J. H. Verrette [12] 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.

Another approach is given by Hambleton and Hausmann [4] for classifying equivariant principal bundles. They used the local invariants arising from isotropy representations at singular points of $(B,\Gamma)$, in the isotropy representation, at a $\Gamma$ fixed point $b_{0}\in B$ it means the homomorphism $\alpha_{b_{0}}:\Gamma\rightarrow G$ defined by the formula

[TABLE]

where $e_{0}\in p^{-1}(b_{0}).$ Denote the collection of isotropy representation of $\xi$ by $Rep^{G}_{\Gamma}(\mathcal{I}).$ The homomorphism $\alpha$ is independent of the choice of $e_{0}$ up to conjugation in $G.$

In this article, $\Gamma$-equivariant principal $G$ bundles over $S^{2}$ will be classified by second approach. First of all, let $X\subset S^{2}$ be a $\Gamma$-invariant $1$-skeleton. The classification of $\Gamma$-equivariant principal $G$-bundle over $X$ will be determined by using the paper of Hambleton and Hausmann [4] since $X$ is a split $\Gamma$-space. On the other hand, $S^{2}$ is not a split $\Gamma$-space. Therefore, we will use a different method to classify $\Gamma$-equivariant principal $G$-bundles over $S^{2}.$ Let $X\subset S^{2}$ be a $\Gamma$-equivariant $CW$- complex, then

[TABLE]

is a cofibration sequence where $CX$= cone on $X$ = $(X\times[0,1])/{(a,0)\sim pt}$ and $\Sigma(S^{2})=\text{suspension of}S^{2}.$

If we get homotopy classes of maps into space $B(\Gamma,G)$,then the following sequence

[TABLE]

is a exact sequence of abelian groups since $B(\Gamma,G)$=$Y=\Omega Z$ is a loop space [2]. We will determine $[S^{2},Y]$ by $\Sigma i^{*}$ and $i^{*}$ since $\Sigma i^{*}$ is the suspension map of $i^{*}.$

However, the classification of principal $G$-bundles over the 2-sphere follows from the Steenrod equivalence theorem [9]. 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.

Theorem 1.1**.**

Let $G$ be a compact connected Lie group, $\Gamma\subset SO(3)$ be a finite subgroup acting linearly on $S^{2}.$ A $\Gamma$-equivariant principal $G$-bundle over $(S^{2},\Gamma)$ is classified by $Rep_{\Gamma}^{G}(\mathcal{I})$ and $c(\xi)\in\pi_{2}(BG).$

Corollary 1.2**.**

If $Rep_{\Gamma}^{G}(\xi_{1})\cong Rep_{\Gamma}^{G}(\xi_{2})$ then $c(\xi_{1})\equiv c(\xi_{2})$ $mod\lvert\Gamma\lvert.$

2 Preliminaries

In this section, we give some definitions and theorems from the book of T. Tom Dieck [11] 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 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 free if for every $x$ $\in$ $X$ the only element of $G$ (identity) fixes the element $x.$ A simplicial $G$-complex is 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 singular points set of $X$ which is the isotropy subgroup of $x$ different than identity is denoted by $Sing(X,G)=\{x\in X\lvert\ G_{x}\neq 1\}.$ 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 $G_{x}S$ = $S$, $S$ is a slice at x if the map

[TABLE]

is a a tube about $G$ $\cdot$ $x.$

Equivalently, we say that a slice at x is defined as a subspace $S$ of $X$ with the following properties:

1. (i)

$S$ is closed in $G$ $\cdot$ $S$ 2. (ii)

$G\cdot S$ is an open neighborhood of $G\cdot x$ 3. (iii)

$G_{x}\cdot S=S$ 4. (iv)

If $(g\cdot S)$ $\cap$ $S$ is non-empty, then $g$ is an element of $G_{x}$

The following theorem says that there exists a slice under certain conditions. It was proven by Montgomery-Yang and Mostow [1957].

Theorem 2.1**.**

[1]** Let $\Gamma$ be a compact Lie group and $X$ be a $\Gamma$-invariant space which is completely regular. Then there is a slice at each point and a tube around each orbit.

After slicing theorem, the 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.

Proposition 2.2**.**

Let $X$ be a compact connected regular simplicial graph, $G$ be a finite group and $G$ $\times$ $X$ $\rightarrow$ $X$ be a regular group action with the orbit space $X$/$G$ $\cong$ $A$ Then

[TABLE]

2.1 Split Spaces

We introduce the definition of a split $\Gamma$-space since Hambleton and Hausmann [4] basically show that if $X$ is split $\Gamma$-space over $A$, then there exists a bijection between the equivalence classes of split $\Gamma$-equivariant G-bundles over $X$ and $Rep^{G}(\mathcal{I})$.

Definition 2.3** (Split $\Gamma$-Space).**

Let $\Gamma$ be a topological group and $A$ be a topological space. A $\Gamma$-space is a topological space equipped with a continuous left action of $\Gamma$. If the space $X$ is a $\Gamma$-space and $x\in X,$ we denote $\Gamma_{x}$ as the stabilizer of $x.$ A * split $\Gamma$-space * over $A$ is denoted by a triple $(X,\pi,\varphi)$ where

• •

$X$* is a $\Gamma$-space.*

• •

$\pi:X\rightarrow A$* is a continuous surjective map and, for each $a\in A$, the preimage $\pi^{-1}(a)$ is a single orbit.*

• •

$\varphi$: A $\rightarrow$ X is a continuous section of $\pi$

In this definition, we might prefer a * split $\Gamma$-space * over $A$ by omitting the notation of maps $\pi$ and $\varphi$. Also, the map $\pi$ induces $\bar{\pi}$: $\Gamma$ $\backslash$ $X$ $\rightarrow$ $A$ which is a homeomorphism since $\varphi$ provides its continuous inverse.

3 Classifying Equivariant Principal G-Bundle

3.1 Split Equivariant Principal Bundles

Now, we introduce the isotropy groupoid for given split $\Gamma$-space. Then we define the isotropy representation of an isotropy groupoid. Finally, we give the theorem which is proven by Hambleton and Hausmann in [4].

Definition 3.1** (Isotropy Groupoid).**

Let (X,$\pi$,$\varphi$) be a split $\Gamma$-space over the space A. The Isotropy groupoid of $X$ is denoted by

[TABLE]

The isotropy groupoid of the space $X$ is the subspace of $\Gamma\times A$ such that for each $a\in A$, the space $\mathfrak{I}_{a}=\mathfrak{I}\cap(\Gamma\times\{a\})$ can be written as the form $\mathfrak{I}_{a}^{\prime}\times\{a\},$ where $\mathfrak{I}_{a}^{\prime}$ is a closed subgroup of $\Gamma.$

We denote by $A^{(n)}$ the skelata of $A,$ by $\Omega$ = $\Omega(A)$ the set of cells of A, by $d(e)$ the dimension of a cell $e$ $\in$ $\Omega$ and $\Omega_{n}$ = $\{$ $e$ $\in$ $\Omega$ $\lvert$ $d(e)$ = $n$ $\}$ for the CW-complex $A.$ Also, we denote by $e(a)$ $\in$ $\Omega$ the cell $e$ of the smallest dimension such that $a$ $\in$ $A.$

Definition 3.2**.**

A $(\Gamma,A)$ groupoid $\mathfrak{I}$ is called cellular if it is locally maximal and if $\tilde{\mathfrak{I_{a}}}=\tilde{\mathfrak{I_{b}}}$ when $e(a)=e(b).$

Definition 3.3** (Split Bundle).**

Let $(X,\pi,\varphi)$ be a split $\Gamma$-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 split bundle if $\varphi^{*}\xi$ is trivial.

A topological space is called a 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.

Any $\Gamma$-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 [4] 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 $S^{1}.$

Definition 3.4** (Isotropy Representation).**

Let $G$ be a topological group and $\mathfrak{I}$ be a $(\Gamma,A)$ groupoid. A continuous representation of $\mathfrak{I}$ is continuous map

[TABLE]

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 $\iota_{a}$: $\mathfrak{I}_{a}$ $\rightarrow$ $G$.

A continuous representation of $\iota$ : $\mathfrak{I}$ $\rightarrow$ $G$ is called * 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 $\alpha_{u}(\gamma)=g(u)\alpha_{a}(\gamma)g(u)^{-1}$ for all $u\in U$ and $\gamma\in\mathfrak{I}_{u}.$ The set of conjugacy classes of locally maximal continuous representations of $\mathfrak{I}$ can be denoted $Rep^{G}(\mathfrak{I}).$

Let $(X,\pi,\varphi)$ be a split $\Gamma$-space over the space A with isotropy groupoid $\mathfrak{I}.$ Let $\xi=(p:E\rightarrow X)$ be a split $\Gamma$-equivariant principal $G$-bundle over the space $X.$ There exists a continuous lifting $\tilde{\varphi}^{*}(\xi):A\rightarrow E$ of $\varphi$ since $\varphi^{*}(\xi)$ is trivial. The equation

[TABLE]

(valid for $a\in A$ and $\gamma\in\mathfrak{I_{a}})$ determines a continuous representation $\alpha_{\xi,\tilde{\varphi}}:\mathfrak{I}\rightarrow G.$ Hambleton and Hausmann [4] checked that $\alpha_{\xi}$ does not depend on the choices $\tilde{\varphi}$ and depends only on the $\Gamma$-equivariant isomorphism class of $\xi$. Then, the classification theorem is the following.

Theorem 3.5** (Classification Theorem of Hambleton and Hausmann).**

[4]** Let $(X,\pi,\varphi)$ be a split $\Gamma$-space over the orbit space $A$ with the isotropy groupoid $\mathfrak{I}$. Assume that $A$ is locally compact, the group $\Gamma$ is a compact Lie group and $\mathfrak{I}$ is locally maximal. Then for any compact Lie group G, the map

[TABLE]

is a bijection.

The $S^{2}$ is the CW-complex but not a split $\Gamma$-space. For this reason, we investigate the restriction of $S^{2}.$ Let $X\subset S^{2}$ be a $\Gamma$-equivariant 1-$skeleton.$ The theorem 3.5 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.

Proposition 3.6**.**

[4]** Let $\Gamma$ be a compact Lie group and let $(X,\pi,\varphi)$ be a split $\Gamma$-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

[TABLE]

3.2 1-skeletons of $S^{2}$

After related definitions and theorems, we will introduce the first theorem of the classification of equivariant bundles over $2$-sphere. If the $\Gamma$-equivariant 1-skeleton is chosen for finite subgroups of $SO(3)$, then it can be concluded that every $\Gamma$-invariant 1-skeleton on $S^{2}$ is split $\Gamma$-space. Hence, if $X$ is a split $\Gamma$-space, then equivariant principal $G$-bundles over the CW-complex $X$ is classified by the theorem 3.5.

Theorem 3.7**.**

Let $\Gamma\subset SO(3)$ be a subgroup acting on $S^{2}$ and $X$ be a $\Gamma$-invariant 1-skeleton on $S^{2}$. Then $X\subset S^{2}$ is a split $\Gamma$-space over $A$.

Proof.

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.

1. (i)

Cyclic Subgroups

Let * $\mathfrak{C_{n}}$* be a 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 * $\mathfrak{C_{n}}$ and $E_{n}$ be the orbit space of * $\mathfrak{C_{n}}$* 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. The CW-complex * $\mathfrak{C_{n}}$* is a split $\mathbb{C}_{n}$-space. It is proven by induction. The orbit space $E_{n}$ and the group * $\mathfrak{C_{n}}$* satisfy the * proposition of Riemann-Hurwitz Formula*;

[TABLE]

and the following holds by definition of Euler Characteristic;

[TABLE]

The claim is that

[TABLE]

After we add 1 edge to the * $\mathfrak{C_{n}}$, the new CW-complex is the * $\mathfrak{C_{n+1}}$. 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.

if $n=2$, $\Gamma=\mathbb{C}_{2}$, then \chi(\emph{ \mathfrak{C_{2}}})=0, $\chi(E_{2})=1$ and

[TABLE]

\chi(\emph{ \mathfrak{C_{n+1}}})=1-n since the CW-complex * $\mathfrak{C_{n+1}}$* has $2$ vertices and $n+1$ edges. Therefore, \chi(\emph{ \mathfrak{C_{n+1}}})=\chi(\emph{ \mathfrak{C_{n}}})-1,   * $\mathfrak{C_{n+1}}$* and $E_{n+1}$ satisfies the proposition of Riemann-Hurwitz Formula.

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 $X.$ 2. (ii)

Dihedral Subgroups

Let $\mathfrak{D}_{n}$ be a 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.

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. Let $\mathcal{D}_{n}$ be the orbit space of $\mathfrak{D}_{n}$ under the group action of $D_{2n}$ . 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

• Riemann-Hurwitz Formula*;

[TABLE]

and the following holds by definition of Euler Characteristic;

[TABLE]

The claim is that

[TABLE]

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. If $n=2$,    then $\Gamma=D_{4}$, the CW-complex $\mathfrak{D}_{2}$ is $1$-skeleton on $S^{2}$ such that it has square stating on equator of $S^{2}$. Let the vertices of the square be named 1, 2, 3, 4.

[TABLE]

then

[TABLE]

and

[TABLE]

$\chi(\mathfrak{D}_{n+1})=-2-4n$ since $\mathfrak{D}_{n+1}$ has $2n+4$ vertices and $6n+6$ edges. Therefore,

[TABLE]

$\mathfrak{D}_{n+1}$ and $\mathcal{D}_{n+1}$ satisfy the Riemann-Hurwitz Formula.

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. 3. (iii)

Tetrahedral Subgroup, Octahedral Subgroup, Icosahedral Subgroup

can be calculated by direct computations without induction. Detailed proof can be followed at [13].

∎

Now, we know that for each finite subgroup of $SO(3),$ we can choose a 1-skeleton CW-complex on $S^{2}$ such that it splits over the orbit space $A$. Hambleton and Hausmann [4] 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.

3.3 Classification of G-bundle Over 1-Skeletons On $S^{2}$

Although we know the general theory about the classification of equivariant principal bundle from [4] and [3], 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 $\Gamma\subset SO(3)$ are simpler than the other spaces. In the proof of theorem 3.7, we choose special 1-skeletons for each subgroup of $SO(3),$ such that each of 1-skeletons is $\Gamma$-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 $\Gamma$-equivariant principal-G bundle is a split $\Gamma$-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 $Rep^{G}(\mathcal{I})$ $\times$ $Bun^{G}(A).$

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.

However, we shall separate two cases since one of the orbit spaces is not contractible:

1. (i)

The orbit space A is contractible. 2. (ii)

The orbit space A is not contractible.

3.3.1 Contractible Case

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 [4]. Since, the possible specific spaces $(*$,$\pi$,$\varphi$ $)$ split over its orbit spaces. All the possible orbit spaces are locally compact, all the possible isotropy groupoid are locally maximal and the group $\Gamma$ is a compact Lie group. Thus

[TABLE]

is a bijection for all case where the orbit space is contractible.

3.3.2 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 $(\mathfrak{D}_{n},\pi,\varphi)$ 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

[TABLE]

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

[TABLE]

If the group $G$ is a connected compact Lie group, it follows

[TABLE]

If the group G is not connected but still a compact Lie group, it follows

[TABLE]

Now, we will define some new concepts to can calculate $Rep^{G}(\mathfrak{I})$ easier.

4 Calculation of $Rep^{G}(\mathfrak{I})$

Let $\iota$ : $\mathfrak{I}$ $\rightarrow$ $G$ be an isotropy representation and let $\mathfrak{I}$ be a $(\Gamma$,$A)$-groupoid. The isotropy representation $\iota$ is called cellular if $\iota_{a}=\iota_{b}$ when $e(a)=e(b).$ For each $e\in\Omega(A),$ this defines Hom($\mathfrak{I}_{e}$, $G)$ with face compatibility conditions $\iota_{e}$ = $\iota_{f}$ $\lvert$ $\mathfrak{I}_{e}$ whenever $f$ $\leq$ $e.$ The set of conjugacy classes of cellular representations of $\mathfrak{I}$ into $G$ is denoted by $Rep^{G}_{cell}$ $(\mathfrak{I}).$ 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 $\beta$

[TABLE]

If we pick an element $b_{e}$ from this product, it must satisfy the face compatibility condition. Now, we will define

[TABLE]

and we can replace $\beta$ as a map $\beta$ : $Rep_{cell}^{G}(\mathfrak{I})$ $\rightarrow$ $\overline{Rep}_{cell}^{G}(\mathfrak{I}).$ Then the diagram is commutative,

[TABLE]

when $\mathfrak{I}$ is proper $(\Gamma$,$A)$-groupoid.

Theorem 4.1**.**

Let $X\subset S^{2}$ $A=X/\Gamma$ Let $\Gamma$ $\subset$ $SO(3)$ and G be a topological group. Let $\mathfrak{I}$ be a $(\Gamma$,$A)$-groupoid, where A is an orbit space. Then $\beta$ : $Rep_{cell}^{G}(\mathfrak{I})$ $\rightarrow$ $\overline{Rep}_{cell}^{G}(\mathfrak{I})$ is surjective.

Proof.

For finite subgroup $\Gamma\subset SO(3),$ 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 [3] 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 $v^{\prime}$ 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 $\iota_{v}^{\prime}\in Hom(\mathfrak{I_{v}^{\prime}},G)$ where ${\iota_{v}^{\prime}}_{\lvert}$ = $\iota_{e}.$ 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 $\beta(\iota)=b.$ ∎

Proposition 4.2**.**

Let $\mathfrak{I}$ is a $(\Gamma,A)$-groupoid, where $\Gamma$ $\subset$ $SO(3)$ 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.

Proof.

One can use the classification theorem and the paper of Hambleton and Hausmann [4]. ∎

Theorem 4.3**.**

Let $\mathfrak{I}$ be a proper $(\Gamma,A)$-groupoid with $\Gamma\subset SO(3)$ 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.

Proof.

Let $\tau(\alpha)$ = $\tau(\alpha^{\prime})$ with the two cellular representations. One can see $\alpha$ = $\alpha^{\prime}$ by taking the conjugate of the one of them. For detailed proof one can look [3]. 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. ∎

Now, we investigate $\tau$ and $\beta$ but we have limited information about $\upsilon$. To do this, we induce from the paper of Hambleton and Hausmann [3]. 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 $\pi_{0}(Z_{\kappa_{1}})\backslash\pi_{0}(Z_{\kappa})/\pi_{0}(Z_{\kappa_{2}}).$

5 Classification of Equivariant Bundles on $1$-skeleton

Now, we know the relation between $Bun^{G}_{\Gamma}(X)$ 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 $\Gamma$ structure for $S^{2}.$ We shall consider $\Gamma=C_{n},D_{2n},A_{4},\mathfrak{O},H.$

Theorem 5.1**.**

Let $X$ = * $\mathfrak{C_{n}}$ be a $\Gamma$-equivariant 1-skeleton * over $S^{2}$ with $2$ vertices and $n$ edges , $C_{n}$ be a cyclic group of order n and acting on * $\mathfrak{C_{n}}$ and $E_{n}$ be the orbit space of * $\mathfrak{C_{n}}$* under the group action of $C_{n}$ with isotropy groupoid $\mathfrak{I}_{1}.$ Then, there is a bijection*

[TABLE]

and

[TABLE]

Proof.

$\mathfrak{C_{n}}$ is a split $C_{n}$-space, all equivariant bundles are split bundles. ∎

Theorem 5.2**.**

Let $X=\mathfrak{D}_{n}$ be a 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

[TABLE]

and

[TABLE]

Proof.

$\mathfrak{D}_{n}$ is split $D_{2n}$-space and

[TABLE]

since the group $G$ is a connected compact Lie group, it follows

[TABLE]

∎

Theorem 5.3**.**

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

[TABLE]

and

[TABLE]

Proof.

$\mathfrak{T}$ is a split $A_{4}$-space, all equivariant bundles are split bundle. ∎

Theorem 5.4**.**

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

[TABLE]

and

[TABLE]

Proof.

$\mathcal{C}$ is a split $\mathfrak{O}$-space, all equivariant bundles are split bundle. ∎

Theorem 5.5**.**

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

[TABLE]

and

[TABLE]

Proof.

$\mathcal{I}$ is a split $H$-space, all equivariant bundles are split bundle. ∎

6 Classification of $\Gamma$-$G$ Bundles over $S^{2}$

Let $X\subset S^{2}$ be a $\Gamma$-equivariant $1$-skeleton, then $\Gamma$-equivariant principal $G$-bundles over $X$ are classified by the isotropy representation since $X$ is split $\Gamma$-space. However, for $S^{2}$, we use a different technique to determine $\Gamma$-equivariant principal $G$-bundles over $S^{2}$ since $S^{2}$ is not a split $\Gamma$-space. Now, let $X\subset S^{2}$ be a $\Gamma$-equivariant $1$-skeleton, then

[TABLE]

is the cofibration sequence of $\Gamma$-equivariant $CW$-complexes where $CX$= cone on $X=(X\times[0,1])/{(a,0)\sim pt}$ and $\Sigma(S^{2})$= suspension of $S^{2}=(X\times[-1,1])/{(a,-1)\sim pt,\ (a,1)\sim pt}$

The $\Gamma$-fixed set of homotopy classes of maps into space $B(\Gamma,G)$,then the following sequence

[TABLE]

is the exact sequnce of abelian groups provided that $B(\Gamma,G)=Y=\Omega Z$ is a loop space which is defined by Costenable and Waner at [2]. Now, $[S^{2},Y]$ is determined by $l^{*}$ and $i^{*}.$

Here, $S^{2}\cup CX\simeq\bigvee S^{2}$ (induced from 2-cells) and $\Sigma(X)\simeq\bigvee S^{2}$ (induced from 1-cells) then the exact sequence at 2 will be the following

[TABLE]

[TABLE]

as a $\Gamma$-module

[TABLE]

[TABLE]

as a $\Gamma$-module

[TABLE]

as a $\Gamma$-module provided that the ideal $I=\mathbb{Z}\{(\gamma-1)\lvert\ \gamma\in\Gamma\}\subset\mathbb{Z}\Gamma.$

For the equation (4), 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 $\Gamma$-equivariant 1-skeleton. For the equation (5), 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 $\Gamma$. For the dihedral group, there are two orbits. For other finite subgroups of $SO(3)$ have a single orbit.

Since $k^{*}$ is an injective map, the following map holds;

[TABLE]

or

[TABLE]

and $Coker(k^{*})$ $\simeq$ $\mathbb{Z}\otimes\pi_{1}(G)$ $\simeq$ $[S^{2},B(\Gamma,G)]_{\Gamma}.$

[TABLE]

Now, let $Z$ and $Y$ be two $\Gamma$-space and $f:Z\rightarrow Y$ be continuous. Define $f^{\gamma}(z)=\gamma^{-1}f(\gamma z)$ $f\rightarrow f^{\gamma}$ gives an action of $\Gamma$ on $[Z,Y]$. Then $f=f^{\gamma}$ $\leftrightarrow$ $\gamma f(z)=f(\gamma z)$ $\leftrightarrow$ $f$ is a $\Gamma$-map. Therefore, we shall say the following $[Z,Y]_{\Gamma}=Fix(\Gamma,[Z,Y])$ .

$Fix(\Gamma,I)=\{x\in I\lvert\gamma x=x\}=0$ and we will determine $Fix(\Gamma,\mathbb{Z}\Gamma).$ Let $t\in\Gamma$ be a generator. $\Gamma$ is acting on $\mathbb{Z}\Gamma=\mathbb{Z}\oplus\mathbb{Z}t\oplus\cdots\oplus\mathbb{Z}t^{n-1}$ where $\lvert\Gamma\lvert=n$. The fixed set of $\mathbb{Z}\Gamma$ will be determined by $(a_{0}+a_{1}t+\cdots+a_{n-1}t^{n-1})\gamma=(a_{0}+a_{1}t+\cdots+a_{n-1}t^{n-1})$ $\forall\gamma\in\Gamma$ implies that fixed elements are $\mathbb{Z}(1+t+\cdots+t^{n-1})$. Then, $Fix(\Gamma,\mathbb{Z}\Gamma)=\mathbb{Z}(1\sum_{\gamma\in\Gamma}\gamma).$ Therefore,

[TABLE]

and

[TABLE]

since we have $[S^{2},Y]=\pi_{2}(Y)$ in the sequence (2).

Theorem 6.1**.**

Let $G$ be a compact Lie group, $\Gamma\subset SO(3)$ finite and $\xi$ be a principal G-bundle over $S^{2}$. $\Gamma$-equivariant principal $G$-bundle over $(S^{2},\Gamma)$ is classified by $Rep_{\Gamma}^{G}(\xi)$ and $c(\xi)\in\pi_{2}(BG).$

Proof.

Let $[\nu]$ and $[\xi]$ $\in$ $[S^{2},Y]_{\Gamma}.$ $[S^{2},Y]_{\Gamma}\xrightarrow{i^{*}}[X,Y]_{\Gamma}$ and $[X,Y]_{\Gamma}\cong Rep^{G}_{\Gamma}(\mathfrak{I})$ If $Rep^{G}_{\Gamma}(\nu)\ncong Rep^{G}_{\Gamma}(\xi)$ then one concludes that they are non-equivariant to each other. if $Rep^{G}_{\Gamma}(\nu)\cong Rep^{G}_{\Gamma}(\xi)$ then

[TABLE]

and by (6) and (7)

[TABLE]

then we have they will be determined by first Chern class. ∎

Corollary 6.2**.**

If $Rep_{\Gamma}^{G}(\xi_{1})\cong Rep_{\Gamma}^{G}(\xi_{2})$ then $c(\xi_{1})\equiv c(\xi_{2})$ $mod\ \lvert\Gamma\lvert.$

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.

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.