# When a free action gives rise to a $G$-principal bundle

When a free action gives rise to a $$G$$-principal bundle

Let a (topological) group $$G$$ act freely on a (topological) space $$X$$. Assume that
$$G \backslash X$$ is Hausdorff. (equivalently the image of the map $$G \times X \to X\times X$$ is closed).

Does it mean that $$X \to G \backslash X$$ is a $$G$$-principal bundle?

We are interested in the case when both $$G$$ and $$X$$ are l-spaces and moreover when $$G=\mathbb Z^n$$. However, I'll be happy to hear about other contexts too: locally compact spaces, topological manifolds, $$C^\infty$$-manifolds, algebraic varieties, etc.

• The answer will depend on your definition of a principal bundle. Section 2 in Chapter 4 of Husemoller's "Fibre bundles", 3rd edition addresses your question for Husemoller's definition. Also see mathoverflow.net/questions/57015/…. Commented May 18, 2020 at 18:52
• So you are not quite asking for the action to be proper? This implies $C\times X\to X\times X$ is a closed map, not just that the image is closed. Commented May 18, 2020 at 21:28
• In the case of $C^\infty$ manifolds a smooth Lie group action on a manifold $X$ is the principal action of a (unique) principal bundle structure $X \to X/G$ iff $G$ acts freely and properly. This seems to be folklore and can be found e.g. in Duistermaat-Kolk "Lie groups" in the first chapter. Commented May 19, 2020 at 15:32
• See here. Commented May 6, 2021 at 12:20

For general topological spaces, I found some answers in Tammo to Dieck, "Algebraic Topology" [TD], Chapter 14.

I give here a quick summary.

Define a principal G-bundle $$p$$ as a continuous map $$p: E \to X$$ and a continuous right group action $$R$$ of $$G$$ on $$E$$ such that:

1. $$p$$ is $$G$$-invariant, that is $$p \circ R_g = p$$
2. $$p$$ is locally trivial in a G-equivariant way (or simply $$G$$-trivial), that is for each $$x \in X$$ there exist a open neighbourhood $$U$$ and a $$G$$-equivariant homeomorphisms $$\varphi: p^{-1}(U) \to U \times G$$ over $$U$$

From now on, let $$E$$ be a space on which $$G$$ acts freely.

Let $$\theta: E \times G \to E \times E, (e,g) \mapsto (e,eg)$$. Let $$C(E)$$ be the image of $$\theta$$. Denote by $$\theta'$$ the restriction of $$\theta$$ onto its image. We call $$t: C(E) \to G, (e,eg) \mapsto g$$ the translation map.

We say the action is weakly proper if $$\theta'$$ is an homeomorphism, or, equivalently, if $$t$$ is continuous. We call the action proper if in addition $$C(E)$$ is closed in $$E \times E$$.

A first observation is the following:

Lemma. If $$p: E \to E/G$$ is locally trivial, then the action is weakly proper

We would like to have a converse of this. Apparently, the action being (free and) weakly proper is not enough for $$E \to E/G$$ to be locally trivial.

But weak properness is used to arrive to a "local triviality condition" on the $$G$$-space $$E$$.

Weak properness gives the usual section-trivialization correspondence for the orbit map:

Proposition. If the action is free and weakly proper, then $$p: E \to E/G$$ is trivial iff $$p$$ has a section

Thus to express (local) triviality we can use the (local) weak properness and the existence of (local) section(s).

To streamline this further, it is proved that

Proposition. For a general $$G$$-space $$E$$ the following are equivalent:

1. There exist a $$G$$-equivariant map $$E \to G$$
2. The orbit map $$E \to E/G$$ is locally $$G$$-trivial
3. The $$G$$-action is free, weakly proper and there exist a (global) section

Main Result

Here it comes the key definition:

Definition. A $$G$$-space $$E$$ it is said to be trivial if it exist a continuous $$G$$-equivariant map $$E \to G$$. $$E$$ is said to be locally trivial if it is covered by trivial subspaces

It follows that:

Proposition. If a $$G$$-space is locally trivial, then $$p:E \to E/G$$ is a $$G$$-principal bundle

Covering spaces

My curiosity was motivated by the analogy with covering spaces. Coverings are fiber bundles with discrete fiber. It is well know that an covering space[*] action of a discrete group $$G$$ gives a $$G$$-principal covering (also called regular or normal covering).

If coverings are fiber bundles with discrete fiber, what is the correct notion (N) on a topological group action $$G$$, such that, when $$G$$ is a discrete group (considered with discrete topology), (N) is equivalent to the condition of a covering space action?

Of course the above discussion gives an answer to this question.

More can be said (also found in [TD]):

If $$G$$ is discrete, the action is a covering space action iff it is free and weakly proper

Unfortunately, as discussed above, if $$G$$ is not discrete, being free and weakly proper it is not enough to ensure local triviality of the orbit map.

[*] Also called properly discontinuous or even. See here on the nomenclature