Diffeomorphic Lift

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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 = φπ.

Proof. By definition of quotient diffeology, the result is locally true, that is for any x ∈ G there exists an open neighbourhood Ux and a smooth map φx : Ux → G such that π ◦ φx = φ ◦ π on Ux. We can suppose that Ux 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, φx(x)h) for some h ∈ G, let

D(x,g) = {(v,(Rh ◦ φx)∗x(v)): v ∈ TxG} ⊂ 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 Ux ∩ Uy ≠ ∅. Then for any z ∈ Ux ∩ Uy and any connected neighbourhood Vz ⊂ Ux ∩ Uy, the local lifts φx, φy define the continuous map γ : Vz → Γ given by γ(t) = φx(t)−1φy(t). Since Vz is connected and the set Γ is totally disconnected, the map γ must be constant, hence φy = Rγ ◦ φx.

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 x0 ∈ G such that [x0] = φ([e]), and let G~ be the maximal integral submanifold passing through (e, x0). 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 p1: G~ ⊂ G × G → G is a covering map.

Proof. Clearly p1 is a differentiable submersion, hence an open map, so p1(G) is an open subspace of G. Let us prove that it is closed too; this will show that the map p1 : G~ → G is onto, because the Lie group G is connected.

Suppose x ∈ G is in the closure of p1(G), and let Ux be a connected open neighbourhood where the local lift φx is defined. Let y ∈ Ux ∩ p1(G), then (y, φx(y)h) ∈ G for some h ∈ G. This implies that the graph of Rh ◦ φx, which is an integral submanifold of D, is contained in G . Hence x ∈ p1(G).

It remains to prove that any x ∈ G has a neighbourhood Ux such that (p1)−1(Ux) is a disjoint union of open sets, each one homeomorphic to Ux by p1. It is clear that we can restrict ourselves to the case x = e. Let φe : Ue → G be a connected local lift of φ. We can suppose that φe(e) = x0.

Let U~e be its graph. Then U~e is an open subset of G~, containing (e, x0), with p1(U~e) = Ue. Let I be the non-empty set

I = {γ ∈Γ : (e,x0γ) ∈ G~} .

Then (p1)−1(U) is the disjoint union of the sets Rγ(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 Rn, 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.

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