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  "body": "\n## Definition\n\nLet $X$ be a nonempty topological space with singular chain complex $C_\\bullet(X)$.\nThe *augmentation* is the homomorphism $\\varepsilon \\colon C_0(X) \\to \\Z$ given by $\\varepsilon\\left(\\sum_i n_i \\sigma_i\\right) = \\sum_i n_i$.\nThe *augmented chain complex* is\n$$\\cdots \\to C_2(X) \\xrightarrow{\\partial_2} C_1(X) \\xrightarrow{\\partial_1} C_0(X) \\xrightarrow{\\varepsilon} \\Z \\to 0,$$\nwith $\\Z$ placed in degree $-1$; it is a chain complex because $\\varepsilon \\circ \\partial_1 = 0$.\nThe *reduced homology* $\\tilde H_n(X)$ is the homology of the augmented chain complex in degree $n$.\n\n## Remarks\n\nFor $n \\ge 1$, $\\tilde H_n(X) = H_n(X)$, where $H_n$ is [[singular homology|M0Q5CFJ7ZH]].\nIn degree $0$, $H_0(X) \\cong \\tilde H_0(X) \\oplus \\Z$, and $\\tilde H_{-1}(X) = 0$ because $\\varepsilon$ is onto.\nFor any point $x_0 \\in X$, $\\tilde H_n(X) \\cong H_n(X, \\set{x_0})$ for all $n$, a [[relative homology|4BVWH0BN28]] group.\n\nExample: for a one-point space $X$, $C_n(X) \\cong \\Z$ for every $n \\ge 0$, and $\\varepsilon$ is an isomorphism, so $\\tilde H_n(X) = 0$ for all $n$.\n\nNon-example: the hypothesis that $X$ is nonempty matters.\nFor $X = \\emptyset$ the map $\\varepsilon \\colon 0 \\to \\Z$ is not onto, so the augmented complex gives $\\tilde H_{-1}(\\emptyset) = \\Z$, and the formula $H_0 \\cong \\tilde H_0 \\oplus \\Z$ fails.\n",
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