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  "body": "\n## Statement\n\nLet $n \\ge 5$.\nLet $W$ be a compact smooth $(n+1)$-manifold whose boundary is a disjoint union $\\partial W = M_0 \\sqcup M_1$ of two closed $n$-manifolds.\nAssume that $W$ is an *h-cobordism*: both inclusions $M_0 \\hookrightarrow W$ and $M_1 \\hookrightarrow W$ are [[homotopy equivalences|XZG6MTSWCN]].\nAssume also that $M_0$ is connected and [[simply connected|S1XK3HXHFX]].\nThen $W$ is [[diffeomorphic|0KZZ70ZHXP]] to $M_0 \\times [0, 1]$ by a diffeomorphism that is the identity on $M_0$.\nIn particular $M_0$ and $M_1$ are diffeomorphic.\n\n## Remarks\n\nThe theorem is due to Smale.\nIdea of the proof: a Morse function on $W$ gives a [[handle decomposition|9V11N8VVTP]] of $W$ on $M_0$.\nThe homotopy equivalence condition makes the handle chain complex acyclic, and handles are traded and cancelled in pairs until none remain.\nCancelling a pair needs the Whitney trick to remove extra intersection points of spheres in a level set.\nThe Whitney trick needs simple connectivity and level sets of dimension at least $5$.\nA cobordism with no handles is a product.\n\nThe theorem at work: let $\\Sigma$ be a smooth closed $m$-manifold homotopy equivalent to $S^m$ with $m \\ge 6$.\nRemoving two disjoint open balls from $\\Sigma$ gives an h-cobordism between two copies of $S^{m-1}$, which is therefore a product.\nIt follows that $\\Sigma$ is homeomorphic to $S^m$, the [[generalized Poincaré conjecture|8KDV43WZ3Z]] in these dimensions.\n\nBoth extra hypotheses are needed.\nWithout simple connectivity there are h-cobordisms that are not products; the obstruction is the Whitehead torsion, see the [[s-cobordism theorem|E1E5ABP89W]].\nFor $n = 4$ the smooth statement fails: Donaldson found smooth $5$-dimensional h-cobordisms between simply connected $4$-manifolds that are not products.\n\n## Explanations\n\n### L1 human\n\nTwo manifolds are h-cobordant when they form the two ends of a bigger manifold that each end fills up, up to homotopy.\nThe theorem says that in high dimensions, when there are no loops to worry about, such a bigger manifold is just a cylinder.\nSo the two ends are the same manifold.\nIn this way the theorem proves the Poincaré conjecture in dimensions at least six.\n\n### L2 human\n\nThe h-cobordism theorem converts a homotopy-theoretic condition into a diffeomorphism.\nIt is the base of the classification of high-dimensional manifolds through [[surgery theory|MHHN8386TG]]: surgery produces cobordisms, and the theorem shows that suitable cobordisms are products.\nIts proof is the model application of Morse theory and handle cancellation.\nThe [[s-cobordism theorem|E1E5ABP89W]] extends it to manifolds with a nontrivial fundamental group.\nIn dimension $4$ the topological version holds by Freedman's work, but the smooth version fails.\n",
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