# Contents

## Definition

Given a topological group or algebraic group or Lie group, etc., $G$, a homogeneous $G$-space is a topological space or scheme, or smooth manifold etc. with transitive $G$-action.

A principal homogeneous $G$-space is the total space of a $G$-torsor over a point.

There are generalizations, e.g. the quantum homogeneous space for the case of quantum groups.

## Examples

• A special case of homogeneous spaces are coset spaces arising from the quotient $G/H$ of a group $G$ by a subgroup. For the case of Lie groups this is also called Klein geometry.

• Specifically for $G$ a compact Lie group and $T\hookrightarrow G$ a maximal torsu?, then the coset $G/T$ play a central role in representation theory and cohomology, for instance in the splitting principle.

## Properties

Under weak topological conditions (cf. Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces), every topological homogeneous space $M$ is isomorphic to a coset space $G/H$ for a closed subgroup $H\subset G$ (the stabilizer of a fixed point in $X$).

Revised on March 29, 2014 03:51:53 by Urs Schreiber (185.37.147.12)