This is a follow up to The 4-sphere does not admit dimension 2 foliations , where I asked about the existence of nonsingular foliations of a 4-sphere.
Since that question determined there are no such foliations, I am now looking for examples of any 2-dimensional foliations of the 4-sphere. The simplier the better; for instance the 3-sphere has a foliation coming from the Hopf map which one can explicitly describe with coordinates. I have been struggling to find something similar for the 4-sphere case and haven't had any luck.
Does anyone know of any?
I will try and answer my own question, since I have been looking at it and have a few more ideas. To get dimension 2 foliations of a 4-sphere, I will start with (singular) 3-dimensional foliations. If each leaf of the foliation has a 2-dimensional foliation, then we will have a dimension 2 foliation of $\mathbb{S}^4$.
Looking at an article by Scardua and Seade in "Foliations, Geometry, and Toplogy", they give several examples of 3-dimensional foliations of the 4-sphere.
Leaves which are $\mathbb{S}^3$ with 2 isolated centers.
Leaves which are $\mathbb{S}^1\times \mathbb{S}^2$ and a singular set which is two circles.
First foliate the 3-sphere with the Hopf fibration, with singular set a Hopf link. Then (I think) the first case gives us a foliation of $\mathbb{S}^4$ with the singular set as two copies of the Hopf links. In the second case we foliate $\mathbb{S}^1\times\mathbb{S}^2$ with 2-spheres with 2 isolated centers, so that should foliate $\mathbb{S}^4$ as 2-spheres with 4 circles for the singular set (I guess they should be linked, but I don't know how).
The other example I was thinking of are Cappell-Shaneson manifolds. These are homotopy 4-spheres which are topologically $\mathbb{R}\times T^3/(t,x)\sim(t-1,\phi(x))$ under a diffeomorphism $\phi$ of the 3-torus $T^3$. Recent results by Akbulut and Gompf have shown that in some cases these guys are also diffeomorphic to the 4-sphere. Then we can foliate the 3-torus with 2-tori as leaves to get a foliaton of the 4-sphere. I have no idea what the singular set would be in this case.
Anyone have any input?