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        "quote": "第 $r$ 个 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:b536c648-0767-49e9-b712-1849d0825566\" data-concept-route=\"/concept-identities#concept-b536c648-0767-49e9-b712-1849d0825566\" id=\"first-57c2dbd73bffb785e74e-zh\">德拉姆上同调(de Rham cohomology)</u> 空间（群）定义为商空间 $H ^ { r } ( M ) = Z ^ { r } ( M ) / B ^ { r } ( M )$，并且任意两个 $r$ 上闭链 $\\alpha$ 与 $\\beta$ 称为 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:114ca9e4-e0ac-482e-9731-3186bbc7dbfa\" data-concept-route=\"/concept-identities#concept-114ca9e4-e0ac-482e-9731-3186bbc7dbfa\" id=\"first-9f5e2943e091be4de6eb-zh\">上同调的(cohomologous)</u>，如果它们属于同一个陪集，即对某个 $(r-1)$ 上链 $\\gamma$ 有 ${ \\alpha - \\beta = { \\mathrm { d } } \\gamma }$。向量空间 $H ^ { r } ( M )$ 的维数记为 $b ^ { r }$。",
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        "quote": "The $r$th <u class=\"concept-term\" data-concept-identity=\"urn:uuid:b536c648-0767-49e9-b712-1849d0825566\" data-concept-route=\"/concept-identities#concept-b536c648-0767-49e9-b712-1849d0825566\" id=\"first-57c2dbd73bffb785e74e-en\">de Rham cohomology</u> space (group) is defined as the factor space $H ^ { r } ( M ) = Z ^ { r } ( M ) / B ^ { r } ( M )$ , and any two $r$-cocycles $\\alpha$ and $\\beta$ are said to be <u class=\"concept-term\" data-concept-identity=\"urn:uuid:114ca9e4-e0ac-482e-9731-3186bbc7dbfa\" data-concept-route=\"/concept-identities#concept-114ca9e4-e0ac-482e-9731-3186bbc7dbfa\" id=\"first-9f5e2943e091be4de6eb-en\">cohomologous</u> if they belong to the same coset, ${ \\alpha - \\beta = { \\mathrm { d } } \\gamma }$ for some $(r-1)$-cochain $\\gamma$ . The dimensions of the vector spaces $H ^ { r } ( M )$ are denoted $b ^ { r }$ .",
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