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  "body": "\n## Summary\n\nReduced homology is a small change of [[singular homology|M0Q5CFJ7ZH]] that makes the homology of a point vanish in every degree.\nIt differs from ordinary homology only in degree $0$, where one copy of $\\Z$ is removed.\nMany statements, such as the homology of spheres, the suspension isomorphism and Alexander duality, take their cleanest form in reduced homology.\n\n## Explanations\n\n### L1 human\n\nOrdinary homology always sees at least one copy of $\\Z$ in degree $0$ for a nonempty space, because the space has at least one piece.\nReduced homology throws this copy away, so that it only records how far the space is from being a single point.\nFor a space made of three separate points, $H_0$ is $\\Z^3$, while the reduced group $\\tilde H_0$ is $\\Z^2$.\nFor the sphere $S^n$, the reduced homology is $\\Z$ in degree $n$ and zero in all other degrees, with no special case for $n = 0$.\n",
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