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 latter denotes the group of all bijections of $X$. If we consider $G$ as being, in some sense, the automorphisms of $X$, the transitivity assumption means that it is impossible to distinguish a particular element 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 to the concept of a transitive group action.

Action on cosets.

Let $G$ be a group, a subgroup, and let $G/H$ denote the set of left 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. The corresponding 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 $X$ to have some extra structure, like a topology or a differential structure. Correspondingly, the group of automorphisms is either a continuous 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 $x$. We identify the space of cosets $G/H_x$ with the homogeneous space by means 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 of the 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 $X$ is assumed to be transitive. To show that ${\tau }_x$ is one-to-one, consider two 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 an element of $H_x$. Therefore $a{H}_x$ and $b{H}_x$ are the same coset.

The homogeneous space as a quotient.

Next, let us show that ${\tau }_x$ 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 the point $gax$, as desired.

Finally, let us note that ${\tau }_x$ identifies the point $x\in X$ with the coset of the identity element $e{H}_x$, that is to say, with the subgroup $H_x$ itself. For this reason, the point $x$ is often called the 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 the quotient structure of a homogeneous space. Let $X$ be a homogeneous space.

Proposition 4

The set of all isotropy subgroups $\mathrm{\{}{H}_x\mathrm{:}x\mathrm{\in }X\mathrm{\}}$ forms a single 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 every possible $x\in X$, it stands to reason that there exist equivariant bijections 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 $h$ is precisely the subgroup $H_1$. Hence, ${\tau }_{x_1}:G/H_1\to G/H_0$ is the desired equivariant bijection. This is a well defined mapping from the set of $H_1$-cosets to the set of $H_0$-cosets, with action given 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 the corresponding coset $G$-actions.

Proposition 5

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

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

Title	homogeneous space
Canonical name	HomogeneousSpace
Date of creation	2013-03-22 13:28:07
Last modified on	2013-03-22 13:28:07
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	6
Author	rmilson (146)
Entry type	Definition
Classification	msc 20A05
Defines	action on cosets
Defines	isotropy subgroup