Contents

group theory

# Contents

## Idea

A common condition on subgroups of a topological group is that they be closed.

This is typically required in equivariant differential topology/equivariant homotopy theory, for instance in the definition of the orbit category.

## Definition

###### Definition

(closed subgroup)

A topological subgroup $H \subset G$ of a topological group $G$ is called a closed subgroup if as a topological subspace it is a closed subspace.

###### Lemma

(open subgroups of topological groups are closed)

Every open subgroup $H \subset G$ of a topological group is closed, hence a closed subgroup.

###### Proof

The set of $H$-cosets is a cover of $G$ by disjoint open subsets. One of these cosets is $H$ itself and hence it is the complement of the union of the other cosets, hence the complement of an open subspace, hence closed.

## Examples

### Stabilizer subgroup

###### Proposition

(stabilizer subgroups of continuous actions on T1-spaces are closed) Let $G$ be a topological group and let $X \in G Actions(TopologicalSpaces)$ a topological G-space whose underlying topological space is a T1-space, e.g. a Hausdorff space.

Then for all points $x \in X$ their isotropy groups, hence their stabilizer subgroup

$G_x \;\coloneqq\; Stab_G(x) \;\subset\; G \,,$

is a closed subgroup (Def. ).

###### Proof

We may understand $G_x$ as the pre-image

$G_x \;\simeq\; \big( \rho(-)(x) \big)^{-1} \big( \{x\} \big)$

of the singleton subset $\{x\} \subset X$ under the continuous function which sends $x$ to its image under the given $G$-action:

$\array{ G &\overset{\rho(-)(x)}{\longrightarrow}& X \\ g &\mapsto& g \cdot x \,. }$

Since this is a continuous function, and since $x \in X$ is a closed point by assumption that $X$ is T1, hence since $\{x\} \subset X$ is a closed subset, it follows that $G_x \subset G$ is a closed subset, since continuous preimages of closed subsets are closed.

### Localic subgroups

Any localic subgroup of a localic group is closed (see this Theorem).

## Properties

### For Lie groups

###### Proposition

(Cartan's closed subgroup theorem)

If $H \subset G$ is a closed subgroup of a (finite dimensional) Lie group, then $H$ is a sub-Lie group, hence a smooth submanifold such that its group operations are smooth functions with respect to the the submanifold smooth structure.

### Local sections over coset space

###### Proposition

(coset space coprojections with local sections)
Let $G$ be a topological group and $H \subset G$ a subgroup.

Sufficient conditions for the coset space coprojection $G \overset{q}{\to} G/H$ to admit local sections, in that there is an open cover $\underset{i \in I}{\sqcup}U_i \to G/H$ and a continuous section $\sigma_{\mathcal{U}}$ of the pullback of $q$ to the cover,

$\array{ && G_{\vert \mathcal{U}} &\longrightarrow& G \\ & {}^{\mathllap{ \exists \sigma }} \nearrow & \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \mathllap{ \exists \; } \underset{i \in I}{\sqcup} U_i &=& \underset{i \in I}{\sqcup} U_i &\longrightarrow& G/H \mathrlap{\,,} }$

include the following:

or:

• The underlying topological space of $G$ is

(e.g. if $G$ is a Lie group)

and $H \subset G$ is a closed subgroup.

## References

Last revised on September 4, 2021 at 14:16:22. See the history of this page for a list of all contributions to it.