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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer and let $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]].\nThe space $X$ is a *homology $n$-manifold* if, for every point $x \\in X$, the local [[relative homology|4BVWH0BN28]] groups satisfy\n$$H_i(X, X \\setminus \\set{x}; \\Z) \\cong \\begin{cases} \\Z \u0026 \\text{if } i = n, \\\\ 0 \u0026 \\text{if } i \\neq n. \\end{cases}$$\n\n## Remarks\n\nSeveral variants are in use: some authors also ask that $X$ be finite-dimensional or an absolute neighbourhood retract, or use sheaf-theoretic homology or coefficients in a ring other than $\\Z$.\n\nExample: a [[topological $n$-manifold|QQQ9B1DK4P]] $M$ is a homology $n$-manifold.\nBy excision, $H_i(M, M \\setminus \\set{x}) \\cong H_i(\\R^n, \\R^n \\setminus \\set{0})$, and the long exact sequence of the pair gives $H_i(\\R^n, \\R^n \\setminus \\set{0}) \\cong \\tilde{H}_{i-1}(S^{n-1})$, which is $\\Z$ for $i = n$ and $0$ otherwise.\n\nNon-example: let $X$ be the wedge of two circles and $x$ the wedge point.\nThen $X \\setminus \\set{x}$ is a disjoint union of two open arcs, and the long exact sequence gives $H_1(X, X \\setminus \\set{x}) \\cong \\Z^3$, so $X$ is not a homology $1$-manifold.\n\nThe suspension of the Poincaré homology sphere is a homology $4$-manifold that is not a topological manifold.\nThe reason is that a small neighbourhood of a suspension point, with that point removed, is not simply connected.\n",
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