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  "body": "\n## Statement\n\nLet $X$ be a closed connected [[topological manifold|QQQ9B1DK4P]] of dimension $n \\ge 5$, with [[fundamental group|S1XK3HXHFX]] $\\pi$ and orientation character $w \\colon \\pi \\to \\set{\\pm 1}$.\nConsider [[simple homotopy equivalences|AR8SP73SXX]] $f \\colon M \\to X$ from closed topological manifolds $M$.\nIdentify $f$ with $f' \\colon M' \\to X$ when there is a homeomorphism $h \\colon M \\to M'$ with $f' \\circ h$ homotopic to $f$.\nThe *structure set* $\\mathcal{S}(X)$ is the set of classes.\nLet $\\mathcal{N}(X)$ be the set of normal bordism classes of degree one normal maps to $X$.\nLet $\\mathcal{N}(X \\times [0,1], \\partial)$ be the same set for $X \\times [0,1]$ relative to the boundary.\nLet $L_m(\\Z[\\pi], w)$ be the simple surgery obstruction groups of Wall.\nThen there is an exact sequence of pointed sets:\n$$\\mathcal{N}(X \\times [0,1], \\partial) \\xrightarrow{\\ \\theta\\ } L_{n+1}(\\Z[\\pi], w) \\xrightarrow{\\ \\partial\\ } \\mathcal{S}(X) \\xrightarrow{\\ \\eta\\ } \\mathcal{N}(X) \\xrightarrow{\\ \\theta\\ } L_n(\\Z[\\pi], w).$$\nHere $\\eta$ sends a homotopy equivalence to its normal invariant and $\\theta$ is the surgery obstruction.\nThe map $\\partial$ comes from an action of $L_{n+1}(\\Z[\\pi], w)$ on $\\mathcal{S}(X)$.\nExactness at $\\mathcal{S}(X)$ means the following.\nTwo elements have the same image under $\\eta$ if and only if they lie in the same orbit of this action.\n\n## Remarks\n\nIdea of the proof: surgeries below the middle dimension make a degree one normal map to $X$ highly connected.\nBy Wall's theorem, the remaining obstruction to reach a homotopy equivalence is $\\theta$.\nThe action of $L_{n+1}$ realises an element as the obstruction of a normal bordism from a given homotopy equivalence; the [[s-cobordism theorem|E1E5ABP89W]] turns trivial obstructions into homeomorphisms.\nIn the topological category $\\mathcal{N}(X)$ is in bijection with the set $[X, G/\\mathrm{TOP}]$ of homotopy classes of maps from $X$ to the space $G/\\mathrm{TOP}$.\n\nThe sequence at work: for $X = S^n$ with $n \\ge 5$, the structure set is a single point, which is the topological generalized Poincaré conjecture.\nFor simply connected $X$ the groups $L_n(\\Z)$ are $\\Z, 0, \\Z/2, 0$ for $n \\equiv 0, 1, 2, 3 \\pmod 4$, and the obstruction in $L_{4k}(\\Z) = \\Z$ is a multiple of a difference of signatures.\n\n## Explanations\n\n### L1 human\n\nSuppose a space looks like a manifold to homotopy theory.\nHow many genuinely different manifolds have this homotopy type?\nThe surgery exact sequence answers this by fitting the set of such manifolds between two computable pieces.\nOne piece is homotopy theory of a classifying space, the other is algebra of quadratic forms over the group ring of the fundamental group.\n\n### L2 human\n\nThe sequence is the main tool of the Browder–Novikov–Sullivan–Wall surgery theory.\nTo classify manifolds homotopy equivalent to $X$, one computes the normal invariants, the surgery obstruction map and the action of $L_{n+1}$.\nIt is the setting of the [[Borel conjecture|X057MN9YY9]], which says that for aspherical $X$ the structure set is a single point.\nThe algebraic theory of these groups is L-theory.\n",
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