homogeneous space



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

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

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



Relation to coset spaces

Under weak topological conditions (cf. Helgason), every topological homogeneous space MM is isomorphic to a coset space G/HG/H for a closed subgroup HGH\subset G (the stabilizer of a fixed point in XX).


Textbook accounts:

On homogeneous spaces with the same rational cohomology as a product of n-spheres:

The following article has categorical analysis of relation between the total space of a principal bundle and of the corresponding quotient space both for the classical case and for noncommutative generalizations: