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  "body": "\n## Summary\n\nThe cap product combines a homology class of degree $n$ and a cohomology class of degree $k \\le n$ into a homology class of degree $n - k$.\nIt makes [[singular homology|M0Q5CFJ7ZH]] a module over the [[cohomology ring|RDT4785M4D]].\nIts main use is Poincaré duality, where capping with the fundamental class of a closed oriented manifold turns cohomology into homology.\n\n## Explanations\n\n### L1 human\n\nA cohomology class is a test on simplices; a homology class is built from simplices.\nThe cap product uses the test on the front part of each simplex and keeps the back part as a smaller simplex.\nSo a degree $k$ test eats $k$ dimensions of an $n$-dimensional chain and leaves an $(n - k)$-dimensional chain.\nWhen $k = n$, nothing but a point is left, and the cap product just records the value of the test on the chain.\n",
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