A compact holonomy-free codimension-one foliation is fibered over its leaf space

(Leaf-space charts and local trivializations.) Let $L_0=q^{-1}(p)$ be a leaf. By [F1] it has a saturated product neighbourhood $U\cong L_0\times D$ with $D$ an open interval, and $U$ is a union of leaves, so $q(U)$ is an open subset of $X$ homeomorphic to $D$: the map $q{\mathrm{\mid }}_U$ is the projection $L_0\times D\to D$ followed by the identification $q(U)\cong D$. These charts make $X$ locally Euclidean of dimension one, and the transition maps between two such charts are the transverse coordinate changes of the foliation, hence homeomorphisms.

(Hausdorffness.) Let $p_1\mathrm{\ne }{p}_2$ in $X$ correspond to distinct leaves $L_1\mathrm{\ne }{L}_2$. These are disjoint compact subsets of the Hausdorff manifold $M$; choosing saturated product neighbourhoods as in [F1] inside disjoint open neighbourhoods of $L_1$ and $L_2$ gives disjoint open sets $q(U_1)\ni p_1$ and $q(U_2)\ni p_2$ in $X$, because a leaf meeting $U_i$ is contained in $U_i$. Hence $X$ is Hausdorff.

(Compactness, connectedness, no boundary.) Since $M$ is nonempty, its quotient $X$ is nonempty. $X$ is compact and connected as a continuous image of $M$ [F2], and by step 1.1 every point of $X$ has a neighbourhood homeomorphic to an open interval, so $X$ has no boundary. Compactness gives finitely many such interval charts covering $X$; the union of their rational-interval bases is a countable base for $X$. Thus $X$ also satisfies the second-countability clause of the manifold definition, and [F3] identifies $X$ with $S^1$.

(Whole-fibre return.) The maps in step 1.1 are local trivializations with fibre $L_0\cong L$. By F4 their interval coordinate changes are smooth, so $X$ is a smooth circle. Choose a positive base vector field of period one, lift it in the finitely many product trivializations, and patch the lifts with a finite smooth partition of unity. The patched field still projects to the base field. Compactness of $M$ gives its flow for time one, and that flow restricts to a diffeomorphism of the whole fibre $L_0$ onto itself. Flow over $[0,1]$ trivializes the pullback bundle; the endpoint gluing is exactly this return map. Convex interpolation of two such lifts, followed by smooth flow dependence, proves that their return maps are isotopic. A closed transversal is a single curve and does not itself determine a whole-fibre return map.

Therefore the leaf space is a compact connected Hausdorff one-manifold homeomorphic to $S^1$ and $q:M\to X$ is a locally trivial fibre bundle with fibre $L$ whose monodromy is the whole-fibre return map for a chosen transverse connection, with the foliation as its fibre foliation.