Given a group $G$ and a subgroup $H$, then their coset is the quotient $G/H$, hence the set of equivalence classes of elements of $G$ where two are regarded as equivalent if they differ by right multiplication with an element in $H$.
If $G$ 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.
In a category $C$, for $G$ a group object and $H \hookrightarrow G$ a subgroup object, the left/right object of cosets is the object of orbits of $G$ under left/right multiplication by $H$.
Explicitly, the left coset space $G/H$ coequalizes the parallel morphisms
where $\mu$ is (the inclusion $H\times G \hookrightarrow G\times G$ composed with) the group multiplication.
Simiarly, the right coset space $H\backslash G$ coequalizes the parallel morphisms
Specializing the above definition to the case where $C$ is the well-pointed topos $Set$, given an element $g$ of $G$, its orbit $gH$ is an element of $G/H$ and is called a left coset.
Using comprehension, we can write
Similarly there is a coset on the right $H \backslash G$.
If $H \hookrightarrow G$ is an inclusion of Lie groups then the quotient $G/H$ is also called a Klein geometry.
More generally, given an (∞,1)-topos $\mathbf{H}$ and a homomorphism of ∞-group ojects $H \to G$, hence equivalently a morphism of their deloopings $\mathbf{B}H \to \mathbf{B}G$, then the homotopy quotient $G/H$ 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.
The coset inherits the structure of a group if $H$ is a normal subgroup.
Unless $G$ is abelian, considering both left and right coset spaces provide different information.
For $X$ a smooth manifold and $G$ a compact Lie group equipped with a free smooth action on $X$, then the quotient projection
is a $G$-principal bundle (hence in particular a Serre fibration).
This is originally due to (Gleason 50). See e.g. (Cohen, theorem 1.3)
For $G$ a Lie group and $H \subset G$ a compact subgroup, then the coset quotient projection
is an $H$-principal bundle (hence in particular a Serre fibration).
This is originally due to (Samelson 41).
For $G$ a compact Lie group and $K \subset H \subset G$ closed subgroups, then the projection map
is a locally trivial $H/K$-fiber bundle (hence in particular a Serre fibration).
Observe that the projection map in question is equivalently
(where on the left we form the Cartesian product and then divide out the diagonal action by $H$). This exhibits it as the $H/K$-fiber bundle associated to the $H$-principal bundle of corollary 1.
In geometric homotopy theory (in an (∞,1)-topos), for $H \longrightarrow G$ any homomorphisms of ∞-group objects, then the natural projection $G \longrightarrow G/H$, generally realizes $G$ as an $H$-principal ∞-bundle over $G/H$. This is exhibited by a homotopy pullback of the form
where $\mathbf{B}H$ is the delooping groupoid of $H$. This also equivalently exhibits the ∞-action of $H$ on $G$ (see there for more).
By the pasting law for homotopy pullbacks then we get the homotopy pullback
which exhibits the coset as the homotopy fiber of $\mathbf{B}H \to \mathbf{B}G$.
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 $\mathbb{R}^{n+1}$. Then its orbit under the defining $O(n+1)$-action on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.
The second statement follows by the same kind of reasoning:
Clearly $U(n+1)$ acts transitively on the unit sphere $S^{2n+1}$ in $\mathbb{C}^{n+1}$. It remains to see that its stabilizer subgroup of any point on this sphere is $U(n)$. If we take the point with coordinates $(1,0, 0, \cdots,0)$ and regard elements of $U(n+1)$ as matrices, then the stabilizer subgroup consists of matrices of the block diagonal form
where $A \in U(n)$.
Consider $K \hookrightarrow H \hookrightarrow G$ two consecutive group inclusions with their induced coset quotient projections
When $G/K \to G/H$ 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 $O(n+1)/O(n) \simeq S^n$ 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 $q \lt n$ then $\pi_q(S^n) = 0$ 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)