homogeneous space

Overview and definition.

Let G be a group acting transitively on a set X. In other words, we consider a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ:GPerm(X), where the latter denotes the group of all bijections of X. If we consider G as being, in some sense, the automorphismsPlanetmathPlanetmath of X, the transitivity assumptionPlanetmathPlanetmath 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 spacePlanetmathPlanetmath. Indeed, the conceptMathworldPlanetmath of a homogeneous space, is logically equivalent to the concept of a transitive group actionMathworldPlanetmath.

Action on cosets.

Let G be a group, H<G a subgroupMathworldPlanetmathPlanetmath, and let G/H denote the set of left cosetsMathworldPlanetmath, as above. For every gG we consider the mapping ψH(g):G/HG/H with action

Proposition 1

The mapping ψH(g) is a bijection. The corresponding mapping ψH:GPerm(G/H) is a group homomorphism, specifying a transitive group action of G on G/H.

Thus, G/H has the natural structureMathworldPlanetmath 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 topologyMathworldPlanetmath or a differential structure. Correspondingly, the group of automorphisms is either a continuous group or a Lie group. In order for the quotient spaceMathworldPlanetmath 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 xX, the subgroup


consisting of all G-actions that fix x, is called the isotropy subgroup at the basepoint x. We identify the space of cosets G/Hx with the homogeneous space by means of the mapping τx:G/HxX, defined by

Proposition 2

The above mapping is a well-defined bijection.

To show that τx is well defined, let a,bG be members of the same left coset, i.e. there exists an hHx such that b=ah. Consequently


as desired. The mapping τx is onto because the action of G on X is assumed to be transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath. To show that τx is one-to-one, consider two cosets aHx,bHx,a,bG such that ax=bx. It follows that b-1a fixes x, and hence is an element of Hx. Therefore aHx and bHx are the same coset.

The homogeneous space as a quotient.

Next, let us show that τx is equivariant relative to the action of G on X and the action of G on the quotientPlanetmathPlanetmath G/Hx.

Proposition 3

We have that


for all gG.

To prove this, let g,aG be given, and note that


The latter coset corresponds under τx to the point gax, as desired.

Finally, let us note that τx identifies the point xX with the coset of the identity elementMathworldPlanetmath eHx, that is to say, with the subgroup Hx itself. For this reason, the point x is often called the basepoint of the identification τx:G/HxX.

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 {Hx:xX} forms a single conjugacy classMathworldPlanetmathPlanetmath of subgroups in G.

To show this, let x0,x1X be given. By the transitivity of the action we may choose a g^G such that x1=g^x0. Hence, for all hG satisfying hx0=x0, we have


Similarly, for all hHx1 we have that g^-1hg^ fixes x0. Therefore,


or what is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, for all xX and gG we have



Since we can identify a homogeneous space X with G/Hx for every possible xX, it stands to reason that there exist equivariant bijections between the different G/Hx. To describe these, let H0,H1<G be conjugate subgroups with


for some fixed g^G. Let us set


and let x0 denote the identityPlanetmathPlanetmathPlanetmathPlanetmath coset H0, and x1 the coset g^H0. What is the subgroup of G that fixes x1? In other words, what are all the hG such that


or what is equivalent, all hG such that


The collectionMathworldPlanetmath of all such h is precisely the subgroup H1. Hence, τx1:G/H1G/H0 is the desired equivariant bijection. This is a well defined mapping from the set of H1-cosets to the set of H0-cosets, with action given by


Let ψ0:GPerm(G/H0) and ψ1:GPerm(G/H1) denote the corresponding coset G-actions.

Proposition 5

For all gG we have that

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