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      "value": "$p$-闭链"
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  "scope_note": "§17.4: homology chain with vanishing boundary ∂C = 0 (C ∈ ker ∂), distinct from a permutation cycle.",
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        "quote": "一个$p$-链$C$若没有<u class=\"concept-term\" data-concept-identity=\"urn:uuid:d89858ae-0ae4-44a6-b482-787218d5e179\" data-concept-route=\"/concept-identities#concept-d89858ae-0ae4-44a6-b482-787218d5e179\">边界(boundary)</u>，即$\\partial C = 0 .$，则称其为<u class=\"concept-term\" data-concept-identity=\"urn:uuid:4ce8076e-d99e-40dd-b263-2ba130402d2c\" data-concept-route=\"/concept-identities#concept-4ce8076e-d99e-40dd-b263-2ba130402d2c\" id=\"first-20322b3f8009b543e4b2-homology-zh\">循环(cycle)</u>。若存在一个$(p+1)$-链$C^{\\prime}$使得$C = \\partial C ^ { \\prime }$，则称其为边界。显然，每个边界都是$p$-循环，因为$\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$，但反之未必成立。",
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        "quote": "A $p$-chain $C$ is said to be a <span id=\"first-20322b3f8009b543e4b2-en\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-identity=\"urn:uuid:4ce8076e-d99e-40dd-b263-2ba130402d2c\" data-concept-route=\"/concept-identities#concept-4ce8076e-d99e-40dd-b263-2ba130402d2c\" id=\"first-20322b3f8009b543e4b2-homology-en\">cycle</u> if it has no <u class=\"concept-term\" data-concept-identity=\"urn:uuid:d89858ae-0ae4-44a6-b482-787218d5e179\" data-concept-route=\"/concept-identities#concept-d89858ae-0ae4-44a6-b482-787218d5e179\" id=\"first-ccd810e4a97cf523defc-en\">boundary</u>, $\\partial C = 0 .$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\\prime}$ such that $C = \\partial C ^ { \\prime }$ . Clearly every boundary is a $p$-cycle since $\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$ , but the converse need not be true.",
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