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Let $M$ be a closed oriented manifold, an oriented foliation $F$ is said non-trivial, if $F$ is not fibration of $M$, i.e. there does not exist a closed manifold $B$, such that $M\overset{F}{\to} B$.

Q Is there any book or anything giving the explicit construct such non-trivial foliation.

PS: What I knew is just the Reeb foliation on $S^3$ and $T^n$ with a family of dense lines? Is there any other example (excluding the product of two manifold)?

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    $\begingroup$ A foliation of a $2$-torus by "irrational" line. All the orbits are dense. $\endgroup$ Mar 21, 2019 at 2:36
  • $\begingroup$ There's also a simpler, 2-dimensional version of the Reeb foliation. Foliate the strip $[0,\pi]\times\mathbb R$ in the plane by the curves $y=c+\csc x$ and the vertical edges of the strip. Then identify the two edges of the strip and roll the resulting cylinder up into a torus. $\endgroup$ Mar 21, 2019 at 3:50

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Foliations of surfaces only exist on surfaces of Euler characteristic 0 and they are all built from suspension foliations and Reeb components. (See the book by Hector and Hirsch).

On 3-manifolds there exist many more foliations. Gabai has proved that for every homology class $h\in H_2$ one finds a taut foliation (without Reeb components) with the surface $H$ (that represents $h$) as a leaf. On the other hand, $H$ can be a fiber only if the Poincaré-dual cohomology class $h\in H^1$ has a finitely generated kernel when seen as a homomorphism $h\colon\pi_1\to Z$. (Stallings Theorem)

Gabai‘s construction is by induction: he cuts the manifold along the surface $H$, shows that this process can be continued until one arrives at a product (bordered surface x interval) (which can obviously be foliated) and then reverses the proces while extending the foliation.

So, if you want explicit examples, one might try to start with a product (bordered surface x interval), and follow Gabai’s paper to glue it along some subsets of the boundary to get nontrivial foliations according to Gabai‘s construction.

If you prefer complicated 1-dimensional foliations, you may look at Vogt‘s paper from Publ. IHES 1988, where he constructs a circle foliation of $R^3$. This can obviously not be a fibration.

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  • $\begingroup$ Sir, is this the reference of Gabai's work : FOLIATIONS AND THE TOPOLOGY OF 3-MANIFOLDS. II ? $\endgroup$
    – DLIN
    Apr 5, 2019 at 7:22
  • $\begingroup$ Yes, this is the paper. $\endgroup$
    – ThiKu
    Apr 6, 2019 at 8:33

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