Constructing exotic $\mathbb{R}^4$'s using vector fields on $\mathbb{R}^5$

I was reading a paper of Arnol'd ("Topological Properties of Eigenoscillations in Mathematical Physics") where he gives the following claim (hopefully I am stating it correctly).

One way to produce smooth 4-dimensional manifolds is to take some smooth, non-vanishing vector field $$v$$ on $$\mathbb{R}^5$$. The flow of this vector field defines a smooth $$\mathbb{R}$$-action on $$\mathbb{R}^5$$, and then we can just take the quotient of $$\mathbb{R}^5$$ by this action to produce some smooth 4-manifold $$M$$.

Arnol'd makes the interesting claim that, given any exotic $$\mathbb{R}^4$$ we know about, it can be produced by a careful choice of this vector field).

Could anyone shed some light on the details of this construction?

$$Exotic(\mathbb R^4) \times \mathbb R$$ is diffeomorphic to $$\mathbb R^5$$ as we know that there exits unique smooth structure on $$\mathbb R^5$$ (proved by Stalling A reference for smooth structures on R^n). Now there exists a nice $$\mathbb R$$ action on $$Exotic(\mathbb R^4)\times \mathbb R$$, i.e, translation along the $$\mathbb R$$ axis. And push-forward of this action will generate a smooth action on $$\mathbb R^5$$ which is the one you are looking for.
• It would be nice to see if there exists any explicit construction. Most of the proofs that I know about exotic structures on $\mathbb R^4$ is in some-sense existential. – Anubhav Mukherjee Jun 23 '20 at 2:36