Let G be a connected and simply connected Lie group, Γ ⊂ G an arbitrary totally disconnected subgroup. While it is possible to develop a general theory of fibre bundles and covering spaces in the diffeological setting, we shall directly prove some lifting properties for the quotient map π : G → G/Γ.

Lifting of diffeomorphisms: The factor space G/Γ is endowed with the quotient diffeology, that is the collection of plots α: U → G/Γ which locally lift through π to a smooth map U → G.

Proposition: Any differentiable map (in the diffeological sense) φ^{–} : G/Γ → G/Γ has a sooth lift φ : G → G, that is a C∞ map such that πφ = φ^{–}π.

Proof. By definition of quotient diffeology, the result is locally true, that is for any x ∈ G there exists an open neighbourhood U_{x} and a smooth map φ_{x} : U_{x} → G such that π ◦ φ_{x} = φ^{–} ◦ π on U_{x}. We can suppose that U_{x} is a connected open set.

Now we define an integrable distribution D on G × G in the following way. Since an arbitrary point (x, g) ∈ G × G can be written as (x, φ_{x}(x)h) for some h ∈ G, let

D_{(x,g)} = {(v,(Rh ◦ φ_{x})_{∗x}(v)): v ∈ T_{x}G} ⊂ T_{(x,g)}(G × G).

The distribution D is well defined because two local lifts differ by some translation. In fact, let x, y ∈ G such that U_{x} ∩ U_{y} ≠ ∅. Then for any z ∈ U_{x} ∩ U_{y} and any connected neighbourhood V_{z} ⊂ U_{x} ∩ U_{y}, the local lifts φ_{x}, φ_{y} define the continuous map γ : V_{z} → Γ given by γ(t) = φ_{x}(t)^{−1}φ_{y}(t). Since V_{z} is connected and the set Γ is totally disconnected, the map γ must be constant, hence φ_{y} = R_{γ} ◦ φ_{x}.

Moreover D has constant rank, and it is integrable, the integral submanifolds being translations of the graphs of the local lifts.

Let us choose some point x_{0} ∈ G such that [x_{0}] = φ^{–}([e]), and let G^{~} be the maximal integral submanifold passing through (e, x_{0}). We shall prove that the projection of G ⊂ G × G onto the first factor is a covering map. Since G is simply connected, it follows that G is the graph of a global lift.

Lemma: The projection p_{1}: G^{~} ⊂ G × G → G is a covering map.

Proof. Clearly p_{1} is a differentiable submersion, hence an open map, so p_{1}(G) is an open subspace of G. Let us prove that it is closed too; this will show that the map p_{1} : G^{~} → G is onto, because the Lie group G is connected.

Suppose x ∈ G is in the closure of p_{1}(G), and let U_{x} be a connected open neighbourhood where the local lift φ_{x} is defined. Let y ∈ U_{x} ∩ p_{1}(G), then (y, φ_{x}(y)h) ∈ G for some h ∈ G. This implies that the graph of R_{h} ◦ φ_{x}, which is an integral submanifold of D, is contained in G . Hence x ∈ p_{1}(G).

It remains to prove that any x ∈ G has a neighbourhood U_{x} such that (p_{1})^{−1}(U_{x}) is a disjoint union of open sets, each one homeomorphic to U_{x} by p_{1}. It is clear that we can restrict ourselves to the case x = e. Let φ_{e} : U_{e} → G be a connected local lift of φ^{–}. We can suppose that φ_{e}(e) = x_{0}.

Let U^{~}_{e} be its graph. Then U^{~}_{e} is an open subset of G^{~}, containing (e, x_{0}), with p_{1}(U^{~}_{e}) = U_{e}. Let I be the non-empty set

I = {γ ∈Γ : (e,x_{0}γ) ∈ G^{~}} .

Then (p_{1})^{−1}(U) is the disjoint union of the sets R_{γ}(U^{~}_{e}), γ ∈ I.

Corollary: Any diffeomorphism of G/Γ can be lifted to a diffeomorphism of G.

Corollary: Let U be a connected simply connected open subset of R^{n}, n ≥ 0. Any differentiable map (resp. diffeomorphism) U × G/Γ → U × G/Γ can be lifted to a C∞ map (resp. diffeomorphism) U × G → U × G.