higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
Given a topological group or algebraic group or Lie group, etc., , a homogeneous -space is a topological space or scheme, or smooth manifold etc. with transitive -action.
A principal homogeneous -space is the total space of a -torsor over a point.
There are generalizations, e.g. the quantum homogeneous space for the case of quantum groups.
A special case of homogeneous spaces are coset spaces arising from the quotient of a group by a subgroup. For the case of Lie groups this is also called Klein geometry.
Specifically for a compact Lie group and a maximal torus, then the coset 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.
Under weak topological conditions (cf. Helgason), every topological homogeneous space is isomorphic to a coset space for a closed subgroup (the stabilizer of a fixed point in ).
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:
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:
Last revised on August 21, 2021 at 19:29:34. See the history of this page for a list of all contributions to it.