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  "body": "\n## Summary\n\nThe Eilenberg–Steenrod axioms list the properties that a homology theory on pairs of spaces should have.\nThey are functoriality, homotopy invariance, a long exact sequence for each pair, excision, and a dimension axiom fixing the homology of a point.\n[[Singular homology|M0Q5CFJ7ZH]] satisfies them, and on finite CW pairs any two theories that satisfy them with the same coefficient group agree.\nDropping the dimension axiom gives generalized homology and cohomology theories such as K-theory and cobordism.\n\n## Explanations\n\n### L1 human\n\nThere are many ways to build homology: from simplices, from cells, from singular chains.\nThe axioms say which properties really matter, so that one can prove things about all these constructions at once.\nThey act like a contract: any construction that keeps the contract gives the same answer on reasonable spaces.\nFor example, from the axioms alone one can compute that the $n$-sphere has homology only in degrees $0$ and $n$.\n",
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