# 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.

## Properties

### Relation between homogenous spaces and coset spaces

Under weak topological conditions (cf. Hegason), 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 $X$).

## References

• Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces

Revised on March 29, 2016 05:20:56 by Urs Schreiber (195.37.209.180)