nLab closed subgroup

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.

(e.g Arhangel’skii-Tkachenko 08, theorem 1.3.5)

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:

• $G$ is any topological group

  and $H$ is a compact Lie group

  (in particular for $H$ a closed subgroup if $G$ itself is a compact Lie group, since closed subspaces of compact Hausdorff spaces are equivalently compact subspaces).

  (Gleason 50, Thm. 4.1)

or:

• The underlying topological space of $G$ is

  1. locally compact;

  2. separable metric space;

  3. of finite dimension of a separable metric space

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

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

  (Mostert 53, Theorem 3)

Related concepts

• maximal compact subgroup

• locally compact topological group

• compact Lie group

References

• Andrew Gleason, Spaces With a Compact Lie Group of Transformations, Proceedings of the American Mathematical Society Vol. 1, No. 1 (Feb., 1950), pp. 35-43 (jstor:2032430, doi:10.2307/2032430)

• Paul Mostert, Local Cross Sections in Locally Compact Groups, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 (jstor:2032540, doi:10.2307/2032540)

• Alexander Arhangel’skii, Mikhail Tkachenko, Topological Groups and Related Structures, Atlantis Press 2008 (doi:10.2991/978-94-91216-35-0)