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  "body": "\n## Summary\n\nHomology with coefficients in an abelian group $G$ is [[singular homology|M0Q5CFJ7ZH]] computed with chains whose coefficients lie in $G$ instead of $\\Z$.\nCoefficients in a field such as $\\Q$ or $\\Z/p$ give vector spaces, which are easier to handle, and coefficients in $\\Z/2$ remove all sign questions.\nThe universal coefficient theorem recovers the homology with any coefficients from the integral homology.\n\n## Explanations\n\n### L1 human\n\nIn ordinary homology a chain is a sum of simplices with integer multiplicities.\nWith coefficients in $G$ the multiplicities are taken from $G$ instead.\nWith $\\Z/2$ coefficients, one only records whether a simplex is present or not, so orientations no longer matter.\nFor the real projective plane this makes a difference: its top homology is $0$ with integer coefficients but $\\Z/2$ with $\\Z/2$ coefficients.\n",
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