Contents

group theory

# Contents

## Idea

Given a group $G$ and a subgroup $H$, then their coset object 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.

## Definition

### Internal to a general category

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

$H \times G \underoverset{\mu}{proj_G}\rightrightarrows G$

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

$G \times H \underoverset{proj_G}{\mu}\rightrightarrows G$

### Internal to $Set$

Specializing the above definition to the case where $C$ is the well-pointed topos $Set$, given an element $g$ of $G$, its orbit $g H$ is an element of $G/H$ and is called a left coset.

Using comprehension, we can write

$G/H = \{g H | g \in G\}$

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

### For Lie groups and Klein geometry

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

### For $\infty$-groups

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

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

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 $H$ is a normal subgroup.

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

### Quotient maps

###### Proposition

For $X$ a smooth manifold and $G$ a compact Lie group equipped with a free smooth action on $X$, then the quotient projection

$X \longrightarrow X/G$

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)

###### Corollary

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

$G \longrightarrow G/H$

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

This is originally due to (Samelson 41).

###### Proposition

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

$p \;\colon\; G/K \longrightarrow G/H$

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

###### Proof

Observe that the projection map in question is equivalently

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

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

### As a homotopy fiber

###### Remark

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

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

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

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

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

## Examples

### $n$-Spheres

###### Example

The n-spheres are coset spaces of orthogonal groups:

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

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

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

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

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

where $A \in U(n)$.

There are also various exceptional realizations of spheres as coset spaces. For instance:

coset space-structures on n-spheres:

standard:
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$this Prop.
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$this Prop.
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$this Prop.
exceptional:
$S^7 \simeq_{diff} Spin(7)/G_2$Spin(7)/G2 is the 7-sphere
$S^7 \simeq_{diff} Spin(6)/SU(3)$since Spin(6) $\simeq$ SU(4)
$S^7 \simeq_{diff} Spin(5)/SU(2)$since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
$S^6 \simeq_{diff} G_2/SU(3)$G2/SU(3) is the 6-sphere
$S^15 \simeq_{diff} Spin(9)/Spin(7)$Spin(9)/Spin(7) is the 15-sphere (from FSS 19, 3.4)

### Sequences of coset spaces

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

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

When $G/K \to G/H$ is a Serre fibration, for instance in the situation of prop. (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

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

Consider a sequence of inclusions of orthogonal groups of the form

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

Then by example we have that $O(n+1)/O(n) \simeq S^n$ is the n-sphere and by corollary the quotient map is a Serre fibration. Hence there is a long exact sequence of homotopy groups of the form

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

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

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