A noncompact foliation violating Novikov's compactness conclusions

Let $N_0=T^2\times S^1$ with coordinates $(x,y)$ on the first torus factor and the circle coordinate suppressed, and let $\omega =dy-\lambda dx$ with $\lambda$ irrational. The form is closed and nowhere vanishing, so by the closed-constant-rank-one-forms criterion its kernel is integrable and defines a codimension-one foliation $F_0$ of $N_0$ by Regular foliation atlases, the product structure being the canonical one of Products of smooth manifolds have a canonical product smooth structure.

The leaf through $([x_0],[y_0],[z_0])$ is parametrized intrinsically by $(u,[v])\mapsto ([x_0+u],[y_0+\lambda u],[v])$. Equality of the first two coordinates would give an integer $k$ with $\lambda k$ an integer, so $k=0$ by irrationality. Plaque continuation gives the intrinsic cylinder $\mathbb{R}\times S^1$. F5 proves density of its torus orbit, hence density of the cylinder in $N_0$. No leaf is compact, so none can be the compact boundary leaf of a Reeb component. Thus $F_0$ is Reebless.

Remove a point $p\in N_0$ and set $M=N_0\setminus \{p\}$, $F=F_0{\mathrm{\mid }}_M$. Then $M$ is a noncompact three-manifold and $F$ is a codimension-one foliation of it; a Reeb component of $F$ would be a compact foliated solid torus in $M$ and hence in $N_0$, forcing a compact leaf of the Reebless foliation $F_0$, so $F$ is Reebless.

Let $L$ be the original cylinder containing $p$. The restricted foliation has the connected leaf $L^{\mathrm{\prime }}=L\setminus \{p\}$; there is no leaf of $F$ through the removed point. Choose one intrinsic plaque disk through $p$ in a small foliation box. Dense $L$ may have other plaques in that box, so its full intersection with the box is not asserted to be this disk. Polar coordinates identify $L$ with ${\mathbb{R}}^2\setminus \{0\}$; if $p$ maps to $a\mathrm{\ne }0$, then $L^{\mathrm{\prime }}\cong {\mathbb{R}}^2\setminus \{0,a\}$. Remove small disjoint disks about these two points and cut off the outer end. The resulting compact pair of pants is a deformation retract along the three end collars, so F5 gives free rank two. The map $z\mapsto (z-a)\mathrm{/}\mathrm{\mid }z-a\mathrm{\mid }$ has degree one on a sufficiently small puncture circle, proving that circle essential in $L^{\mathrm{\prime }}$ by F5.

Center ambient foliation coordinates $(u,v,w)$ at $p$, with the chosen plaque $w=0$. Its puncture circle $u^2+v^2={\epsilon }^2$, $w=0$, bounds the upper hemisphere $u^2+v^2+w^2={\epsilon }^2$, $w\ge 0$. This continuous disk avoids $p$ and lies in the coordinate box. The circle is therefore nullhomotopic in $M$ but essential in $L^{\mathrm{\prime }}$ by step 4.1. Inclusion on fundamental groups is noninjective, while $F$ is smooth, cooriented by $\omega$, oriented in the ambient torus and Reebless. This refutes the extension beyond the closed-manifold hypothesis.