homogeneous space

Overview and definition.

Let $G$ be a group acting transitively on a set $X$. In other words,we consider a homomorphism $\varphi :G\to \mathrm{Perm}(X),$ where the latterdenotes the group of all bijections of $X$. If we consider $G$ asbeing, in some sense, the automorphisms of $X$, the transitivityassumption means that it is impossible to distinguish a particularelement of $X$ from any another element. Since the elements of $X$are indistinguishable, we call $X$ a homogeneous space.Indeed, the concept of a homogeneous space, is logically equivalent tothe concept of a transitive group action.

Action on cosets.

Let $G$ be a group, a subgroup, and let $G/H$ denote the set ofleft cosets, as above. For every $g\in G$ we consider the mapping${\psi }_H(g):G/H\to G/H$ with action

$$aH\to gaH,a\in G.$$

Proposition 1

The mapping ${\psi }_H\mathit{}\mathrm{(}g\mathrm{)}$ is a bijection. Thecorresponding mapping ${\psi }_H\mathrm{:}G\mathrm{\to }\mathrm{Perm}\mathit{}\mathrm{(}G\mathrm{/}H\mathrm{)}$ is a group homomorphism,specifying a transitive group action of $G$ on $G\mathrm{/}H$.

Thus, $G/H$ has the natural structure of a homogeneous space. Indeed,we shall see that every homogeneous space $X$ is isomorphic to $G/H$,for some subgroup $H$.

N.B. In geometric applications, the want the homogeneous space $X$ tohave some extra structure, like a topology or a differentialstructure. Correspondingly, the group of automorphisms is either acontinuous group or a Lie group. In order for the quotient space $X$to have a Hausdorff topology, we need to assume that the subgroup $H$is closed in $G$.

The isotropy subgroup and the basepoint identification.

Let $X$ be a homogeneous space. For $x\in X$, the subgroup

$$H_x=\{h\in G:hx=x\},$$

consisting of all $G$-actions that fix$x$, is called the isotropy subgroup at the basepoint $x$. Weidentify the space of cosets $G/H_x$ with the homogeneous space bymeans of the mapping ${\tau }_x:G/H_x\to X$, defined by

$${\tau }_x(a{H}_x)=ax,a\in G.$$

Proposition 2

The above mapping is a well-defined bijection.

To show that ${\tau }_x$ is well defined, let $a,b\in G$ be members ofthe same left coset, i.e. there exists an $h\in H_x$ such that$b=ah$. Consequently

$$bx=a(hx)=ax,$$

as desired. The mapping ${\tau }_x$ is onto because the action of$G$ on $X$ isassumed to be transitive. To show that ${\tau }_x$ is one-to-one, considertwo cosets $a{H}_x,b{H}_x,a,b\in G$ such that$ax=bx$. It follows that $b^{-1}a$ fixes $x$, and hence is anelement of $H_x$. Therefore $a{H}_x$ and $b{H}_x$ are the same coset.

The homogeneous space as a quotient.

Next, let us showthat ${\tau }_x$ is equivariant relative to the action of $G$ on $X$and the action of $G$ on the quotient $G/H_x$.

Proposition 3

We have that

$$\varphi (g)\circ {\tau }_x={\tau }_x\circ {\psi }_{H_x}(g)$$

for all $g\mathrm{\in }G$.

To prove this, let $g,a\in G$ be given, and note that

$${\psi }_{H_x}(g)(a{H}_x)=ga{H}_x.$$

The latter coset corresponds under ${\tau }_x$ to thepoint $gax$, as desired.

Finally, let us note that ${\tau }_x$ identifies the point $x\in X$ withthe coset of the identity element $e{H}_x$, that is to say, with thesubgroup $H_x$ itself. For this reason, the point $x$ is often calledthe basepoint of the identification ${\tau }_x:G/H_x\to X$.

The choice of basepoint.

Next, we consider the effect of the choice of basepoint on thequotient structure of a homogeneous space. Let $X$ be a homogeneousspace.

Proposition 4

The set of all isotropy subgroups $\mathrm{\{}{H}_x\mathrm{:}x\mathrm{\in }X\mathrm{\}}$ forms asingle conjugacy class of subgroups in $G$.

To show this, let $x_0,x_1\in X$ be given. By the transitivity of the action we may choose a$\widehat{g}\in G$ such that $x_1=\widehat{g}{x}_0$. Hence, for all $h\in G$satisfying $h{x}_0=x_0$, we have

$$(\widehat{g}h{\widehat{g}}^{-1})x_1=\widehat{g}(h({\widehat{g}}^{-1}{x}_1))=\widehat{g}{x}_0=x_1.$$

Similarly, for all $h\in H_{x_1}$ we have that ${\widehat{g}}^{-1}h\widehat{g}$ fixes $x_0$.Therefore,

$$\widehat{g}(H_{x_0}){\widehat{g}}^{-1}=H_{x_1};$$

or what is equivalent, for all $x\in X$ and $g\in G$ we have

$$g{H}_x{g}^{-1}=H_{gx}.$$

Equivariance.

Since we can identify a homogeneous space $X$ with $G/H_x$ for everypossible $x\in X$, it stands to reason that there exist equivariantbijections between the different $G/H_x$. To describe these, let be conjugate subgroups with

$$H_1=\widehat{g}{H}_0{\widehat{g}}^{-1}$$

for some fixed $\widehat{g}\in G$. Let us set

$$X=G/H_0,$$

and let $x_0$ denote the identity coset $H_0$, and $x_1$the coset $\widehat{g}{H}_0$. What is the subgroup of $G$ that fixes $x_1$?In other words, what are all the $h\in G$ such that

$$h\widehat{g}{H}_0=\widehat{g}{H}_0,$$

or what is equivalent, all $h\in G$ such that

$${\widehat{g}}^{-1}h\widehat{g}\in H_0.$$

The collection of all such $h$ isprecisely the subgroup $H_1$. Hence, ${\tau }_{x_1}:G/H_1\to G/H_0$ isthe desired equivariant bijection. This is a well defined mappingfrom the set of $H_1$-cosets to the set of $H_0$-cosets, with actiongiven by

$${\tau }_{x_1}(a{H}_1)=a\widehat{g}{H}_0,a\in G.$$

Let ${\psi }_0:G\to \mathrm{Perm}(G/H_0)$ and ${\psi }_1:G\to \mathrm{Perm}(G/H_1)$ denote thecorresponding coset $G$-actions.

Proposition 5

For all $g\mathrm{\in }G$ we have that

$${\tau }_{x_1}\circ {\psi }_1(g)={\psi }_0(g)\circ {\tau }_{x_1}.$$