# Taut foliations and closed leaves

EDITED

A codimension-1 foliation $\mathcal{F}$ of a (closed, connected, oriented) 3-manifold is taut if there exists a simple closed curve $\gamma$ that intersects each leaf of $\mathcal{F}$ transversely.

Using work of Thurston, you know that if a taut foliation has closed leaves, they must be homologically non-trivial. But how do you know when a taut foliation admits closed leaves at all? Is there some way to tell, or some class of manifolds which you know support a taut foliations with closed leaves?

Thanks!

I'm assuming you meant homologically non-trivial. There are two very famous theorems of Gabai, which show up in quite a lot of places in 3-dimensional topology for constructing taut foliations (in particular the proofs of properties P and R).

Theorem(Gabai): Let $Y$ be an irreducible 3-manifold, a nonzero $\alpha \in H_2(Y)$, and let $\Sigma$ be a minimal complexity surface representing $\alpha$ with no toroidal or spherical components. Then there exists a taut foliation on $Y$ so that $\Sigma$ is a disjoint union of leaves.

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Theorem(Gabai): Let $S^3_0(K)$ denote 0-surgery on a knot $K$ in $S^3$. Let $\Sigma$ be the closed surface formed by gluing a minimal genus Seifert surface to the surgered in disk. Then there exists a taut foliation of $S^3_0(K)$ with $\Sigma$ as a leaf.

The original reference to these theorems are the three papers Foliations and the Topology of 3-Manifolds (I), II and III.

• In the second theorem, the Seifert surface needs to be minimal genus. Commented Oct 20, 2016 at 16:17

The existence of compact leaves of codimension 1 foliations defined on 3-manifolds is characterized by Novikov see:

https://en.wikipedia.org/wiki/Novikov%27s_compact_leaf_theorem

If you read Novikov paper: (Theorem 6.1) and (Theorem 7.1) imply that if the image of $t(A)\rightarrow \pi_1(M)$ has a non trivial kernel, then the foliation has a compact leaf, here $t(A)$ is the semi-group of homotopy class of tranversal through a point $x$ the leaf $A$. In particular, for a taut foliation one can define the class of the closed transversal in $t(A)$ for any leaf and apply the previous theorems if this class is a non trivial element of the kernel $t(A)\rightarrow \pi_1(M)$.

http://www.mi.ras.ru/~snovikov/23.pdf