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

Question

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.

Answer 1

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

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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

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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.

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