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  "body": "\n## Statement\n\nLet $M$ be a compact, simply connected smooth manifold without boundary, and let $\\Lambda M$ be its [[free loop space|SGGXZ3TBSA]].\nAssume that for some field $F$ the sequence of [[Betti numbers|ST71ZV3FES]] $b_k(\\Lambda M; F) = \\dim_F H_k(\\Lambda M; F)$, $k \\ge 0$, is unbounded.\nThen for every Riemannian metric $g$ on $M$, the [[Riemannian manifold|7CBT4SM83Y]] $(M, g)$ has infinitely many geometrically distinct [[closed geodesics|4MCCP8GG4J]].\n\n## Remarks\n\nThe idea of the proof is this.\nSuppose there were only finitely many geometrically distinct closed geodesics.\nEach critical orbit of the energy on $\\Lambda M$, with all its iterates, contributes to the homology of $\\Lambda M$ only in a controlled way.\nThe indices of iterates grow linearly, and the local homology of iterates takes only finitely many forms.\nThis forces the Betti numbers of $\\Lambda M$ to be bounded, a contradiction.\n\nBy a theorem of Vigué-Poirrier and Sullivan, for $F = \\Q$ the Betti numbers of $\\Lambda M$ are unbounded exactly when the rational cohomology ring $H^*(M; \\Q)$ needs at least two generators.\nSo the hypothesis holds, for example, for $M = S^2 \\times S^2$.\nThe hypothesis fails for spheres and for compact rank one symmetric spaces, whose rational cohomology is generated by one element.\nFor $S^2$ the conclusion still holds by the theorem of [[Bangert and Franks|X93AEE88PX]].\n\n## Explanations\n\n### L1 human\n\nSuppose the space of all loops on a manifold has very complicated topology.\nThen every Riemannian metric on the manifold has infinitely many essentially different closed geodesics.\nThe complexity is measured by the Betti numbers of the loop space, which count holes of each dimension.\nThe point is that finitely many closed geodesics, even with all their repetitions, cannot account for unbounded numbers of holes.\n\n### L2 human\n\nThe theorem is a basic result of Morse theory on the free loop space.\nThere closed geodesics are critical orbits of the energy, for the circle action that rotates the parameter.\nIts main technical inputs are the growth of the Morse index under iteration and a finiteness result for local homology of degenerate critical orbits.\nIt is a well-known open question whether every compact simply connected Riemannian manifold of dimension at least $2$ has infinitely many closed geodesics.\nThe Gromoll–Meyer theorem settles it when the loop space homology grows.\n",
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