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  "body": "\n## Definition\n\nLet $X$ be a topological space and let $G$ be an [[abelian group|082GHNPJ53]].\nThe group $C_n(X; G)$ of singular $n$-chains with coefficients in $G$ is the set of finite formal sums $\\sum_i g_i \\sigma_i$.\nHere $g_i \\in G$ and $\\sigma_i \\colon \\Delta^n \\to X$ are singular $n$-simplices; equivalently, $C_n(X; G) = C_n(X) \\otimes_{\\Z} G$.\nThe boundary map is $\\partial\\left(\\sum_i g_i \\sigma_i\\right) = \\sum_i \\sum_{k=0}^{n} (-1)^k g_i \\, \\sigma_i|_{[v_0, \\ldots, \\hat v_k, \\ldots, v_n]}$, where the hat means that the vertex is omitted.\nThis gives a chain complex, and the *homology of $X$ with coefficients in $G$*, written $H_n(X; G)$, is its homology in degree $n$.\nFor a subspace $A \\subseteq X$, the relative groups $H_n(X, A; G)$ come from $C_n(X; G) / C_n(A; G)$ in the same way.\n\n## Remarks\n\nFor $G = \\Z$ this is ordinary [[singular homology|M0Q5CFJ7ZH]].\nIf $R$ is a commutative ring, then $H_n(X; R)$ is an $R$-module; for a field it is a vector space.\nThe universal coefficient theorem gives a split exact sequence $0 \\to H_n(X) \\otimes G \\to H_n(X; G) \\to \\operatorname{Tor}(H_{n-1}(X), G) \\to 0$, with [[Tor|DDQRERAQCV]] the Tor functor.\n\nExample: for the [[real projective plane|1DDYB6VVSR]], $H_1(\\R P^2) = \\Z/2$ and $H_2(\\R P^2) = 0$.\nThe universal coefficient theorem gives $H_2(\\R P^2; \\Z/2) \\cong 0 \\oplus \\operatorname{Tor}(\\Z/2, \\Z/2) \\cong \\Z/2$.\n\nNon-example: the construction needs $G$ to be abelian, since the boundary formula adds coefficients in an arbitrary order.\nThe proof that $\\partial \\circ \\partial = 0$ rearranges the double sum and cancels terms in pairs, which uses that addition in $G$ is commutative.\n",
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