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Let $M^3$ be a smooth, closed 3-manifold. Given a smooth codimension-one foliation $\mathcal{F}$ of $M$, the Novikov compact leaf theorem asserts that, when the universal cover $\widetilde{M}^3$ of $M^3$ is not contractible, the foliation $\mathcal{F}$ contains a compact leaf of genus $g \leq 1$. What is the most basic example of a smooth codimension-one foliation with all noncompact leaves or leaves of higher genus on a smooth three manifold that has a contractible universal cover, such as the $3$-torus?

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    $\begingroup$ What about $\Sigma_g\times S^1$? $\endgroup$
    – Michael Albanese
    Commented Jun 20, 2020 at 20:26
  • $\begingroup$ Thank you! That answers the question of higher genus, but I am still confused about examples with all noncompact leaves. $\endgroup$
    – Daniel Santiago
    Commented Jun 20, 2020 at 20:46
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    $\begingroup$ Just take the codimension one foliation of $T^2$ by non-compact leaves (of irrational slope), and cross with $S^1$. This gives a codimension one foliation of $T^3$ by $S^1\times\mathbb{R}$. $\endgroup$
    – Michael Albanese
    Commented Jun 20, 2020 at 21:02

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