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  "body": "\n## Statement\n\nLet $n \\ge 5$.\nLet $W$ be a compact connected smooth $(n+1)$-manifold whose boundary is a disjoint union $\\partial W = M_0 \\sqcup M_1$ of closed $n$-manifolds.\nAssume that both inclusions $M_i \\hookrightarrow W$ are homotopy equivalences.\nLet $\\pi = \\pi_1(M_0)$ be the [[fundamental group|S1XK3HXHFX]], let $\\operatorname{Wh}(\\pi)$ be its Whitehead group, and let $\\tau(W, M_0) \\in \\operatorname{Wh}(\\pi)$ be the Whitehead torsion of the inclusion $M_0 \\hookrightarrow W$.\n\n1. $W$ is diffeomorphic to $M_0 \\times [0, 1]$ relative to $M_0$ if and only if $\\tau(W, M_0) = 0$.\n2. For every closed connected $n$-manifold $M_0$ with fundamental group $\\pi$ and every $\\tau \\in \\operatorname{Wh}(\\pi)$, there is such an h-cobordism $W$ with $\\tau(W, M_0) = \\tau$.\n3. Two such h-cobordisms on $M_0$ are diffeomorphic relative to $M_0$ if and only if they have the same torsion.\n\n## Remarks\n\nThe theorem is due independently to Barden, Mazur and Stallings.\nThe same statements hold for PL manifolds, and for topological manifolds by work of Kirby and Siebenmann.\nIdea of the proof: as for the [[h-cobordism theorem|56YF8W9NXP]], take a handle decomposition and cancel handles.\nNow the handle chain complex is a free based complex over $\\Z[\\pi]$, and the obstruction to cancelling all handles is exactly its torsion in $\\operatorname{Wh}(\\pi)$.\n\nThe theorem at work: $\\operatorname{Wh}(1) = 0$, which gives back the h-cobordism theorem, and $\\operatorname{Wh}(\\Z^k) = 0$ by the computation for free abelian groups.\nSo every h-cobordism of dimension at least $6$ on a torus is a product.\nFor $\\pi = \\Z/5$ the Whitehead group is infinite cyclic, so a closed $5$-manifold with this fundamental group carries h-cobordisms that are not products.\n\n## Explanations\n\n### L1 human\n\nThe h-cobordism theorem needs the manifolds to have no nontrivial loops.\nThe s-cobordism theorem drops this hypothesis and says exactly what can go wrong.\nThere is one algebraic invariant, the Whitehead torsion, and the bigger manifold is a cylinder exactly when this invariant vanishes.\n\n### L2 human\n\nThe theorem explains why surgery theory works with simple homotopy equivalences: an h-cobordism with zero torsion is an s-cobordism, and s-cobordant manifolds of dimension at least $5$ are diffeomorphic.\nIt is one input of the [[surgery exact sequence|Q7C480HQ9J]], where the structure set is formed up to this relation.\nWhitehead torsion is computed from algebraic K-theory, since $\\operatorname{Wh}(\\pi)$ is a quotient of $K_1(\\Z[\\pi])$.\nClassical examples come from lens spaces, whose distinction up to simple homotopy uses Reidemeister torsion.\n",
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