nLab homogeneous space

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 torus, then the coset $G/T$ play a central role in representation theory and cohomology, for instance in the splitting principle.

• In analysis and number theory, certain functions on certain coset spaces play a role as automorphic forms (e.g. modular forms). See there for more.

Properties

Relation to coset spaces

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

Related concepts

• Klein geometry

• conjugacy class

• orbit category

• Grassmannian, flag variety

• Schubert calculus

References

Textbook accounts:

• Glen Bredon, Section I.4 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)

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

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

• Linus Kramer, Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface, Memoirs of the American Mathematical Society number 752 (arXiv:math/0109133, doi:10.1090/memo/0752, GoogleBooks)

A category theoretic 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:

• Tomasz Brzeziński, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35 1D (2010) 13-27 [arxiv:0912.0213]

• Tomasz Brzeziński, Quantum group differentials, bundles and gauge theory, Encyclopedia of Mathematical Physics, Acad. Press. (2006) 236-244 [doi:10.1016/B0-12-512666-2/00050-X]

On coset spaces (homogeneous spaces) and their Maurer-Cartan forms in application to first-order formulation of (super-)gravity:

• Leonardo Castellani, L. J. Romans, Nicholas P. Warner, Symmetries of coset spaces and Kaluza-Klein supergravity, Annals of Physics 157 2 (1984) 394-407 [doi:10.1016/0003-4916(84)90066-6]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §I.6 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.6: pdf

• Leonardo Castellani, On G/H geometry and its use in M-theory compactifications, Annals Phys. 287 (2001) 1-13 [arXiv:hep-th/9912277, doi:10.1006/aphy.2000.6097]