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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?

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levitopher
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    The Reeb foliation of $S^{3}$ together with your singular foliation of $S^{4}$ by $S^{3}$s (with two isolated centers) gives a 2-dimensional foliation singular at two points. Naturally, there are lots of examples with one singular point gotten by foliating $\mathbf{R}^{4}$ and viewing $S^{4}$ as the one-point compactification. Presumably you're looking for more than this...? – Andrew D. Hwang Mar 02 '15 at 17:20
  • I am looking for any confirmation that the answer I provided below makes sense, which includes the idea that 1-foliation of $\mathbb{S}^3$ with a foliation of $\mathbb{S}^4$ by $\mathbb{S}^3$s, so your comment is helpful! – levitopher Mar 02 '15 at 18:13

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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?

levitopher
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