# Codimension two foliations with transverse surfaces

Suppose I have some closed $$4$$-manifold $$X$$ and a codimension-two foliation $$\mathcal{F}$$, as well as a closed surface $$\Sigma$$ of nonnegative self-intersection that is everywhere transverse to $$\mathcal{F}$$.

Then what kind of restrictions are there on the foliation $$\mathcal{F}$$? This question gives some answers in the case where $$X$$ is a complex surface and $$\mathcal{F}$$ is holomorphic, but I'm more interested in what happens in the real case.

• For the special case that $\Sigma$ is the fiber of a surface bundle structure for $X$, the foliations you ask about are completely classified. That probably barely scatches the surface of the problem though. Aug 19, 2020 at 0:37
• What is the classification in this case? Aug 19, 2020 at 0:51
• Foliations transverse to the fibering of a fiber bundle with compact fiber, in any dimension and codimension, correspond to conjugacy classes of representations of $\pi_1(B)$, into $Homeo(F)$. Where $B$ is the base space and $F$ is the fiber. Let $\rho$ be such a representation, and let $\tilde{B}$ be the universal cover of $B$ and let $G = \{(\alpha, \rho(\alpha)\}$ where $\alpha$ ranges over $\pi_1(B)$. To construct the foliation, mod out $\tilde{B} \times F$ by $G$. The leaves descend from the product foliation of $\tilde{B} \times \{point\}$, and leafwise cover $B$ under bundle projection. Aug 19, 2020 at 1:48

In this real case, there are few restrictions. Indeed, choose $$\Sigma\subset X$$ such that $$X$$ admits a smooth 2-plane field $$\xi$$ (not necessarily integrable) transverse to $$\Sigma$$. Then, it is easy to perturb slightly $$\xi$$ to make it integrable on a small neighborhood of $$\Sigma$$. Then, by a theorem of Thurston (Commentarii 1974), $$\xi$$, being of real dimension $$2$$, can be homotoped rel. $$\Sigma$$ to become integrable everywhere. You can even begin with extending $$\xi$$ to a partial foliation of your choice over any regular subset of $$X$$. So, the possibilities are enormous.