Given a group GG and a subgroup HH, then their coset object is the quotient G/HG/H, hence the set of equivalence classes of elements of GG where two are regarded as equivalent if they differ by right multiplication with an element in HH.

If GG 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 CC, for GG a group object and HGH \hookrightarrow G a subgroup object, the left/right object of cosets is the object of orbits of GG under left/right multiplication by HH.

Explicitly, the left coset space G/HG/H coequalizes the parallel morphisms

H×Gμproj GG H \times G \underoverset{\mu}{proj_G}\rightrightarrows G

where μ\mu is (the inclusion H×GG×GH\times G \hookrightarrow G\times G composed with) the group multiplication.

Simiarly, the right coset space H\GH\backslash G coequalizes the parallel morphisms

G×Hproj GμG G \times H \underoverset{proj_G}{\mu}\rightrightarrows G

Internal to SetSet

Specializing the above definition to the case where CC is the well-pointed topos SetSet, given an element gg of GG, its orbit gHg H is an element of G/HG/H and is called a left coset.

Using comprehension, we can write

G/H={gH|gG} G/H = \{g H | g \in G\}

Similarly there is a coset on the right H\GH \backslash G.

For Lie groups and Klein geometry

If HGH \hookrightarrow G is an inclusion of Lie groups then the quotient G/HG/H is also called a Klein geometry.

For \infty-groups

More generally, given an (∞,1)-topos H\mathbf{H} and a homomorphism of ∞-group ojects HGH \to G, hence equivalently a morphism of their deloopings BHBG\mathbf{B}H \to \mathbf{B}G, then the homotopy quotient G/HG/H is given by the homotopy fiber of this map

G/H BH BG. \array{ G/H &\longrightarrow& \mathbf{B}H \\ && \downarrow \\ && \mathbf{B}G } \,.

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 HH is a normal subgroup.

Unless GG is abelian, considering both left and right coset spaces provide different information.

Quotient maps


For XX a smooth manifold and GG a compact Lie group equipped with a free smooth action on XX, then the quotient projection

XX/G X \longrightarrow X/G

is a GG-principal bundle (hence in particular a Serre fibration).

This is originally due to (Gleason 50). See e.g. (Cohen, theorem 1.3)


For GG a Lie group and HGH \subset G a compact subgroup, then the coset quotient projection

GG/H G \longrightarrow G/H

is an HH-principal bundle (hence in particular a Serre fibration).

This is originally due to (Samelson 41).


For GG a compact Lie group and KHGK \subset H \subset G closed subgroups, then the projection map

p:G/KG/H p \;\colon\; G/K \longrightarrow G/H

is a locally trivial H/KH/K-fiber bundle (hence in particular a Serre fibration).


Observe that the projection map in question is equivalently

G× H(H/K)G/H, G \times_H (H/K) \longrightarrow G/H \,,

(where on the left we form the Cartesian product and then divide out the diagonal action by HH). This exhibits it as the H/KH/K-fiber bundle associated to the HH-principal bundle of corollary 1.

As a homotopy fiber


In geometric homotopy theory (in an (∞,1)-topos), for HGH \longrightarrow G any homomorphisms of ∞-group objects, then the natural projection GG/HG \longrightarrow G/H, generally realizes GG as an HH-principal ∞-bundle over G/HG/H. This is exhibited by a homotopy pullback of the form

G * G/H BH. \array{ G & \longrightarrow &* \\ \downarrow && \downarrow \\ G/H &\longrightarrow& \mathbf{B}H } \,.

where BH\mathbf{B}H is the delooping groupoid of HH. This also equivalently exhibits the ∞-action of HH on GG (see there for more).

By the pasting law for homotopy pullbacks then we get the homotopy pullback

G/H BH * BG \array{ G/H & \longrightarrow &\mathbf{B}H \\ \downarrow && \downarrow \\ * & \longrightarrow & \mathbf{B}G }

which exhibits the coset as the homotopy fiber of BHBG\mathbf{B}H \to \mathbf{B}G.




The n-spheres are coset spaces of orthogonal groups:

S nO(n+1)/O(n). S^n \simeq O(n+1)/O(n) \,.

The odd-dimensional spheres are also coset spaces of unitary groups:

S 2n+1U(n+1)/U(n) S^{2n+1} \simeq U(n+1)/U(n)

Regarding the first statement:

Fix a unit vector in n+1\mathbb{R}^{n+1}. Then its orbit under the defining O(n+1)O(n+1)-action on n+1\mathbb{R}^{n+1} is clearly the canonical embedding S n n+1S^n \hookrightarrow \mathbb{R}^{n+1}. But precisely the subgroup of O(n+1)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)O(n), hence S nO(n+1)/O(n)S^n \simeq O(n+1)/O(n).

The second statement follows by the same kind of reasoning:

Clearly U(n+1)U(n+1) acts transitively on the unit sphere S 2n+1S^{2n+1} in n+1\mathbb{C}^{n+1}. It remains to see that its stabilizer subgroup of any point on this sphere is U(n)U(n). If we take the point with coordinates (1,0,0,,0)(1,0, 0, \cdots,0) and regard elements of U(n+1)U(n+1) as matrices, then the stabilizer subgroup consists of matrices of the block diagonal form

(1 0 0 A) \left( \array{ 1 & \vec 0 \\ \vec 0 & A } \right)

where AU(n)A \in U(n).

Sequences of coset spaces

Consider KHGK \hookrightarrow H \hookrightarrow G two consecutive group inclusions with their induced coset quotient projections

H/K G/K G/H. \array{ H/K & \longrightarrow& G/K \\ && \downarrow \\ && G/H } \,.

When G/KG/HG/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

π n+1(G/H)π n(H/K)π n(G/K)π n(G/H)π n1(H/K). \cdots \to \pi_{n+1}(G/H) \longrightarrow \pi_n(H/K) \longrightarrow \pi_n(G/K) \longrightarrow \pi_n(G/H) \longrightarrow \pi_{n-1}(H/K) \to \cdots \,.

Consider a sequence of inclusions of orthogonal groups of the form

O(n)O(n+1)O(n+k). O(n) \hookrightarrow O(n+1) \hookrightarrow O(n+k) \,.

Then by example 1 we have that O(n+1)/O(n)S nO(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

π q(S n)π q(O(n+k)/O(n))π q(O(n+k)/O(n+1))π q1(S n). \cdots \to \pi_q(S^n) \longrightarrow \pi_q(O(n+k)/O(n)) \longrightarrow \pi_q(O(n+k)/O(n+1)) \longrightarrow \pi_{q-1}(S^n) \to \cdots \,.

Now for q<nq \lt n then π q(S n)=0\pi_q(S^n) = 0 and hence in this range we have isomorphisms

π <n(O(n+k)/O(n))π <n(O(n+k)/O(n+1)). \pi_{\bullet \lt n}(O(n+k)/O(n)) \stackrel{\simeq}{\longrightarrow} \pi_{\bullet \lt n}(O(n+k)/O(n+1)) \,.


  • 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)

Revised on May 22, 2017 09:46:24 by Urs Schreiber (