Reeb-Thurston stability for codimension-one leaves with vanishing first real cohomology

(${\pi }_1$ and the real cohomology of $L$.) The leaf $L$ is compact and connected, so ${\pi }_1(L,x)$ is finitely generated by [F2], and $L$ is path-connected; [F5] gives $H^1(L;\mathbb{R})\cong \mathrm{Hom}({\pi }_1(L,x),\mathbb{R})$. Thus the hypothesis $H^1(L;\mathbb{R})=0$ says exactly that every homomorphism ${\pi }_1(L,x)\to \mathbb{R}$ is zero.

(Holonomy is trivial.) The holonomy of $L$ is the homomorphism ${\rho }_x:{\pi }_1(L,x)\to {\mathrm{Diff}}_x^{1,+}(T)$ of [F1]. Suppose its image $H$ were nontrivial. Then $H$ is a finitely generated subgroup of the group of $C^1$ germs, by [F3] applied to ${\rho }_x$ and [F2]; by local indicability [F4] there is a surjective homomorphism $H\twoheadrightarrow \mathbb{Z}$. Composing with the inclusion $\mathbb{Z}\hookrightarrow \mathbb{R}$ and with ${\rho }_x$ yields a nonzero homomorphism ${\pi }_1(L,x)\to \mathbb{R}$, that is, by step 1.1 a nonzero element of $H^1(L;\mathbb{R})$, contradicting the hypothesis. Hence $H$ is trivial and the holonomy of $L$ vanishes.

(Product neighbourhood and diffeomorphic leaves.) Since $L$ is compact and has trivial $C^1$ holonomy, [F6] provides a saturated open neighbourhood $U$ of $L$ and a $C^1$ foliated diffeomorphism $U\cong L\times D$ carrying $F{\mathrm{\mid }}_U$ to the product foliation by the slices; in particular every leaf of $F$ meeting $U$ is $C^1$-diffeomorphic to $L$, and $U$ is a union of leaves. This proves the local cohomological stability statement.