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homogeneous space (Definition)

Overview and definition.

Let $ G$ be a group acting transitively on a set $ X$. In other words, we consider a homomorphism $ \phi:G\to\operatorname{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, $ H<G$ 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
$\displaystyle a H \to ga H,\quad a\in G.$
Proposition 1   The mapping $ \psi_H(g)$ is a bijection. The corresponding mapping $ \psi_H:G\to\operatorname{Perm}(G/H)$ is a group homomorphism, specifying a transitive group action of $ G$ on $ G/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
$\displaystyle H_x = \{ h\in G: hx = x \},$
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
$\displaystyle \tau_{x}(aH_x) = ax,\quad a\in G.$
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
$\displaystyle bx = a(hx) = ax,$
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 $ aH_x, bH_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 $ aH_x$ and $ bH_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
$\displaystyle \phi(g)\circ\tau_x = \tau_x\circ\psi_{H_x}(g)$
for all $ g\in G$.
To prove this, let $ g,a\in G$ be given, and note that
$\displaystyle \psi_{H_x}(g)(aH_x)=gaH_x.$
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 $ eH_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 $ \{ H_x : x\in X\}$ 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 $ \hat{g}\in G$ such that $ x_1 = \hat{g}x_0$. Hence, for all $ h\in G$ satisfying $ hx_0 = x_0$, we have
$\displaystyle (\hat{g}h \hat{g}^{-1}) x_1 = \hat{g}(h ( \hat{g}^{-1} x_1)) = \hat{g}x_0 = x_1.$
Similarly, for all $ h\in H_{x_1}$ we have that $ \hat{g}^{-1} h \hat{g}$ fixes $ x_0$. Therefore,
$\displaystyle \hat{g}(H_{x_0}) \hat{g}^{-1} = H_{x_1};$
or what is equivalent, for all $ x\in X$ and $ g\in G$ we have
$\displaystyle g H_x g^{-1} = H_{gx}.$

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 $ H_0, H_1<G$ be conjugate subgroups with
$\displaystyle H_1 = \hat{g}H_0 \hat{g}^{-1}$
for some fixed $ \hat{g}\in G$. Let us set
$\displaystyle X=G/H_0,$
and let $ x_0$ denote the identity coset $ H_0$, and $ x_1$ the coset $ \hat{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
$\displaystyle h \hat{g}H_0 = \hat{g}H_0,$
or what is equivalent, all $ h\in G$ such that
$\displaystyle \hat{g}^{-1} h \hat{g}\in H_0.$
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
$\displaystyle \tau_{x_1}(a H_1)=a \hat{g}H_0,\quad a\in G.$

Let $ \psi_0: G\to \operatorname{Perm}(G/H_0)$ and $ \psi_1:G\to \operatorname{Perm}(G/H_1)$ denote the corresponding coset $ G$-actions.

Proposition 5   For all $ g\in G$ we have that
$\displaystyle \tau_{x_1}\circ\psi_1(g) = \psi_0(g)\circ \tau_{x_1}.$



"homogeneous space" is owned by rmilson.
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Also defines:  action on cosets, isotropy subgroup
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Cross-references: collection, identity, equivalent, conjugacy class, quotient structure, identity element, point, quotient, equivariant, one-to-one, transitive, onto, well-defined, cosets, basepoint, fix, closed, Hausdorff topology, quotient space, order, Lie group, continuous, differential structure, topology, isomorphic, structure, action, mapping, left cosets, subgroup, transitive group action, logically equivalent, transitivity, automorphisms, bijections, homomorphism, words, group
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This is version 3 of homogeneous space, born on 2003-02-14, modified 2003-02-14.
Object id is 4038, canonical name is HomogeneousSpace.
Accessed 6733 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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