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  "body": "\n## Statement\n\nLet $M$ be a closed connected manifold and let $\\mathcal{F}$ be a transversely orientable [[foliation|98HTB7MSPX]] of $M$ of class $C^1$ and codimension $1$.\nAssume that $\\mathcal{F}$ has a [[compact|0TVJZYB6SH]] leaf $L$ whose [[fundamental group|S1XK3HXHFX]] is finite.\nThen every leaf of $\\mathcal{F}$ is compact and diffeomorphic to $L$, and there is a fibre bundle $M \\to S^1$ whose fibres are the leaves of $\\mathcal{F}$.\n\n## Remarks\n\nThe theorem is a global consequence of the local Reeb stability theorem.\nThe local theorem says that a compact leaf with finite holonomy group has a neighbourhood made of leaves that are finite covers of it.\nIdea of the proof: since $\\pi_1(L)$ is finite and its holonomy acts on a transverse interval preserving orientation, the holonomy is trivial, so nearby leaves are copies of $L$.\nThe set of points on compact leaves diffeomorphic to $L$ is then open, and a limit argument shows it is closed, so it is all of $M$.\n\nExample: if a closed $3$-manifold $M$ has a transversely orientable codimension one foliation with a leaf homeomorphic to $S^2$, the theorem shows that $M$ fibres over $S^1$ with fibre $S^2$.\nThe hypothesis on the fundamental group is needed: the [[Reeb foliation|W58NYR1DGR]] of $S^3$ has a compact torus leaf, with infinite fundamental group, while all its other leaves are planes.\n\n## Explanations\n\n### L1 human\n\nA foliation slices a manifold into sheets of one lower dimension.\nThe theorem says that one closed sheet with almost no loops forces all sheets to be copies of it.\nThen the manifold is a stack of these copies arranged around a circle.\n\n### L2 human\n\nThe theorem shows that spheres and other leaves with finite fundamental group are rigid inside codimension one foliations.\nIt is a first step in the study of compact leaves, which leads to Novikov's compact leaf theorem.\nThe key notion is holonomy, the return map of a transverse interval along loops in a leaf, see the holonomy groupoid of a foliation.\nA corollary is that $S^3$ carries no codimension one foliation with a leaf diffeomorphic to $S^2$, since $S^3$ does not fibre over the circle.\n",
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