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  "body": "\n## Summary\n\nA homology $n$-sphere is a closed $n$-manifold with the same integral homology groups as the [[$n$-sphere|T62369RFH6]].\nThe first example that is not a sphere is the Poincaré homology sphere, a $3$-manifold with a nontrivial fundamental group of order $120$.\nHomology spheres show that homology alone does not determine a manifold, and they were central in the history of the Poincaré conjecture.\n\n## Explanations\n\n### L1 human\n\nHomology counts holes of each dimension.\nA homology sphere has exactly the holes of an ordinary sphere and no others.\nStill, it can differ from the sphere: it may contain loops that cannot be shrunk to a point, though each such loop has trivial homology class.\nThe Poincaré homology sphere, obtained by gluing opposite faces of a solid dodecahedron with a twist, is the classical example.\n",
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