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coset

Contents

Idea

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.

Definition

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.

Properties

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

Proposition

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)

Corollary

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).

Proposition

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).

Proof

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

Remark

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.

Examples

nn-Spheres

Example

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)
Proof

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 \,.
Example

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)) \,.

References

  • 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 August 2, 2017 04:09:35 by David Corfield (91.88.197.172)