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  "body": "\n## Definition\n\nLet $X$ be a topological space, let $G$ be an [[abelian group|082GHNPJ53]], and let $C_\\bullet(X)$ be the singular chain complex of $X$ with integer coefficients.\nThe group of *singular $n$-cochains* is $C^n(X; G) = \\Hom(C_n(X), G)$, that is, the group of all functions from singular $n$-simplices of $X$ to $G$.\nThe *coboundary* $\\delta \\colon C^n(X; G) \\to C^{n+1}(X; G)$ is $\\delta \\varphi = \\varphi \\circ \\partial_{n+1}$.\nThen $\\delta \\circ \\delta = 0$.\nThe *singular cohomology* group $H^n(X; G)$ is $\\ker(\\delta \\colon C^n \\to C^{n+1}) / \\im(\\delta \\colon C^{n-1} \\to C^n)$.\nFor a subspace $A \\subseteq X$, the relative group $H^n(X, A; G)$ is defined in the same way from $\\Hom(C_n(X, A), G)$.\nThese are the cochains that vanish on all simplices in $A$.\n\n## Remarks\n\nA continuous map $f \\colon X \\to Y$ induces $f^* \\colon H^n(Y; G) \\to H^n(X; G)$, so $H^n(-; G)$ is a contravariant functor; compare the Hom functor.\nEvaluation $\\langle \\varphi, c \\rangle = \\varphi(c)$ of a cochain on a chain gives the *Kronecker pairing* $H^n(X; G) \\times H_n(X) \\to G$.\nThe universal coefficient theorem gives a split exact sequence $0 \\to \\operatorname{Ext}(H_{n-1}(X), G) \\to H^n(X; G) \\to \\Hom(H_n(X), G) \\to 0$, with [[Ext|4P7GBWJ4RR]] the Ext functor and $H_n$ [[singular homology|M0Q5CFJ7ZH]].\nWhen $G = \\Z$ one writes $H^n(X)$.\n\nExample: a $0$-cochain $\\varphi$ is a function from points of $X$ to $G$, and $(\\delta \\varphi)(\\sigma) = \\varphi(\\sigma(1)) - \\varphi(\\sigma(0))$ for a path $\\sigma$.\nSo $\\varphi$ is a cocycle exactly when it is constant on each path component.\nHence $H^0(X; G)$ is the group of functions from the set of path components of $X$ to $G$.\n\nNon-example: on $X = [0, 1]$ the $0$-cochain $\\varphi(x) = x$ with $G = \\R$ is not a cocycle, because $\\delta\\varphi$ takes the value $1$ on the path $t \\mapsto t$.\n",
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