# 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 |