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        "quote": "A differential $r$-form $\\alpha$ is said to be <u class=\"concept-term\" data-concept-identity=\"urn:uuid:8da45318-5944-4aa7-afd5-245f6894ccb2\" data-concept-route=\"/concept-identities#concept-8da45318-5944-4aa7-afd5-245f6894ccb2\">closed</u> if $\\mathrm{d}\\alpha=0$, and it is said to be <u class=\"concept-term\" data-concept-identity=\"urn:uuid:1a0a5e0c-3f3c-401a-baab-d7dd7829326b\" data-concept-route=\"/concept-identities#concept-1a0a5e0c-3f3c-401a-baab-d7dd7829326b\" id=\"first-b62ff826920c0e8eb620-en\">exact</u> if there exists an $(r-1)$-form $\\beta$ such that $\\alpha = \\mathrm { d } \\beta$ . Clearly every exact $r$-form is closed since $\\mathbf { d } ^ { 2 } { \\boldsymbol { \\beta } } = 0$ . In the language of cochains these definitions can be expressed as follows: an $r$-cochain $\\alpha$ is called an $r$-<u class=\"concept-term\" data-concept-identity=\"urn:uuid:c26a11a5-b8bb-4cd7-8920-cd092b71eb06\" data-concept-route=\"/concept-identities#concept-c26a11a5-b8bb-4cd7-8920-cd092b71eb06\" id=\"first-6a9d37016a96e22681a4-en\">cocycle</u> if it is a closed differential form, while it is an $r$-<u class=\"concept-term\" data-concept-identity=\"urn:uuid:91f394ee-b446-4502-be7d-9d5169bdd21e\" data-concept-route=\"/concept-identities#concept-91f394ee-b446-4502-be7d-9d5169bdd21e\" id=\"first-0b1865b872f69033467f-en\">coboundary</u> if it is exact. We denoted the vector subspace of $r$-cocycles by $Z ^ { r } ( M )$ , and the subspace of $r$-coboundaries by $B ^ { r } ( M )$",
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