Using Godement's criterion to prove that leaf space of a foliation carries a smooth structure compatible with the quotient topology.

Question

I am trying to prove the following from Differential Geometry by Rui Loja Fernandes:

Let $\mathcal{F}$ be a foliation of a smooth manifold $M$. The following statements are equivalent:

1. There exists a smooth structure on $M/\mathcal{F}$, compatible with the quotient topology, such that $π : M → M/\mathcal{F}$ is a submersion.
2. The leaf space $M/\mathcal{F}$ is Hausdorff and there is a cover of $M$ by foliated charts with the property that each leaf of F intersects each chart at most once.

The Book says to use Godement's criterion which says that:

Let $M$ be a smooth manifold and let ∼ be an equivalence relation on $M$. The following statements are equivalent:

1. There exists a smooth structure on $M/ ∼$, compatible with the quotient topology, such that $π : M → M/ ∼$ is a submersion.
2. The graph $R$ of ∼ is a proper submanifold of $M ×M$ and the restriction of the projection $p_1 : M × M → M$ to $R$ is a submersion.

Where $$ R= \{(x,y)\in M\times M; x∼y\}$$

I have shown that 1 implies 2. I am having trouble trying to show that 2 implies 1, in particular, that $R$ is a submanifold. I believe that having done that all of the rest follows straightforwardly from the criterion. Any tips/suggestions? (Also if you know of any other way to show this please let me know !)

Answer 1

I'm going to take my chances and try to answer the question:

First, I think the idea is to use the proof of Godement's Criterion that is written in those notes, and not the result itself.

For 1. implies 2., since we are assuming $\pi$ is a submersion, then the smooth structure on the quotient is actually unique: this is due to the characteristic property of smooth submersion (Theorem 4.29. in Lee's Introduction to Smooth Manifolds). In the proof of Godement's Criterion, in order to build charts $\left(\pi(U), \bar{x}^{k+1}, \ldots, \bar{x}^d\right)$ that define the smooth structure on $M/\mathcal{F}$ one needs to prove that

For every $p \in M$, there exists a local chart $\left(U,\left(x^1, \ldots, x^d\right)\right)$ centered at $p$ such that \begin{align} \forall q, q^{\prime} \in U, q \sim q^{\prime} \text { if and only if } x^{k+1}(q)=x^{k+1}\left(q^{\prime}\right), \ldots, x^d(q)=x^d\left(q^{\prime}\right) \text {. } \end{align}

In this case $q \sim q^{\prime}$ means $q$ and $q^{\prime}$ belong to the same leaf. Since the quotient smooth structure exists by hypothesis and is unique, it must be built by these charts, which are foliated charts in $M$ such that each leaf intersects each chart at most once.

For 2. implies 1., notice that by hypothesis we already have guaranteed the existence of the charts $\left(U,\left(x^1, \ldots, x^d\right)\right)$ with the properties above, so one can follow along the proof of Godement's Criterion and conclude that there exists a smooth structure. As far as I'm aware, the only tricky part is that at a certain point in the proof one must show that the projection to the space of leaves $\pi$ is an open map. I've only found this result in Camacho, Geometric Theory of Foliations, Theorem 1, page 47, but there may exist another reference that I'm not aware of.