Cohomology and Extensions
Given a group and a subgroup , then their coset is the quotient , hence the set of equivalence classes of elements of where two are regarded as equivalent if they differ by right multiplication with an element in .
If is a topological group, then the quotient is a topological space and usually called the coset space. This is in particular a homogeneous space, see there for more.
Internal to a general category
In a category , for a group object and a subgroup object, the left/right object of cosets is the object of orbits of under left/right multiplication by .
Explicitly, the left coset space coequalizes the parallel morphisms
where is (the inclusion composed with) the group multiplication.
Simiarly, the right coset space coequalizes the parallel morphisms
Specializing the above definition to the case where is the well-pointed topos , given an element of , its orbit is an element of and is called a left coset.
Using comprehension, we can write
Similarly there is a coset on the right .
For Lie groups and Klein geometry
If is an inclusion of Lie groups then the quotient is also called a Klein geometry.
More generally, given an (∞,1)-topos and a homomorphism of ∞-group ojects , hence equivalently a morphism of their deloopings , then the homotopy quotient is given by the homotopy fiber of this map
See at ∞-action for more on this definition. See at [[higher Klein geometry] and higher Cartan geometry for the corresponding concepts of higher geometry.
For normal subgroups
The coset inherits the structure of a group if is a normal subgroup.
Unless is abelian, considering both left and right coset spaces provide different information.
This is originally due to (Gleason 50). See e.g. (Cohen, theorem 1.3)
This is originally due to (Samelson 41).
As a homotopy fiber
The n-spheres are coset spaces of orthogonal groups:
The odd-dimensional spheres are also coset spaces of unitary groups:
Regarding the first statement:
Fix a unit vector in . Then its orbit under the defining -action on is clearly the canonical embedding . But precisely the subgroup of that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to , hence .
The second statement follows by the same kind of reasoning:
Clearly acts transitively on the unit sphere in . It remains to see that its stabilizer subgroup of any point on this sphere is . If we take the point with coordinates and regard elements of as matrices, then the stabilizer subgroup consists of matrices of the block diagonal form
Sequences of coset spaces
Consider two consecutive group inclusions with their induced coset quotient projections
When is a Serre fibration, for instance in the situation of prop. 2 (so that this is indeed a homotopy fiber sequence with respect to the classical model structure on topological spaces) then it induces the corresponding long exact sequence of homotopy groups
Consider a sequence of inclusions of orthogonal groups of the form
Then by example 1 we have that is the n-sphere and by corollary 1 the quotient map is a Serre fibration. Hence there is a long exact sequence of homotopy groups of the form
Now for then and hence in this range we have isomorphisms
H. Samelson, Beitrage zur Topologie der Gruppenmannigfaltigkeiten, Ann. of Math. 2, 42, (1941), 1091 - 1137.
Andrew Gleason, Spaces with a compact Lie group of transformations, Proc. of A.M.S 1, (1950), 35 - 43.
Norman Steenrod, section I.7 of The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press, 1951.
R. Cohen, Topology of fiber bundles, Lecture notes (pdf)