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  "body": "\n## Definition\n\nLet $X$ be a [[CW complex|5W95QFHXPR]] with $n$-skeleton $X^n$ for $n \\ge 0$, and put $X^{-1} = \\emptyset$.\nThe *cellular chain group* in degree $n$ is the [[relative homology|4BVWH0BN28]] group $C_n^{\\mathrm{CW}}(X) = H_n(X^n, X^{n-1})$, taken in [[singular homology|M0Q5CFJ7ZH]] with integer coefficients.\nLet $\\partial$ be the connecting map of the pair $(X^n, X^{n-1})$, and let $j$ be induced by the inclusion of pairs.\nThe *cellular boundary map* $d_n \\colon C_n^{\\mathrm{CW}}(X) \\to C_{n-1}^{\\mathrm{CW}}(X)$ is the composite\n$$H_n(X^n, X^{n-1}) \\xrightarrow{\\ \\partial\\ } H_{n-1}(X^{n-1}) \\xrightarrow{\\ j\\ } H_{n-1}(X^{n-1}, X^{n-2}).$$\nThen $d_{n-1} \\circ d_n = 0$, so these groups and maps form a chain complex.\nThe *cellular homology* $H_n^{\\mathrm{CW}}(X)$ is the homology of this chain complex in degree $n$.\n\n## Remarks\n\nThe group $H_n(X^n, X^{n-1})$ is free abelian with one generator $e_\\alpha$ for each $n$-cell $e_\\alpha$ of $X$.\nIn this basis, $d_n(e_\\alpha) = \\sum_\\beta d_{\\alpha\\beta} e_\\beta$, where $d_{\\alpha\\beta}$ is the [[degree|TMFG2P912R]] of the map $S^{n-1} \\to X^{n-1} \\to S^{n-1}$ given by the attaching map of $e_\\alpha$ followed by the collapse of everything outside $e_\\beta$.\nThe basic theorem is that $H_n^{\\mathrm{CW}}(X) \\cong H_n(X)$ for all $n$.\n\nExample: the [[real projective plane|1DDYB6VVSR]] $\\R P^2$ has one cell in each of the dimensions $0$, $1$, $2$.\nThe $2$-cell is attached by a map of degree $2$ onto the $1$-skeleton $S^1$, so $d_2 = 2$, and $d_1 = 0$ because the $1$-cell is a loop.\nThe cellular chain complex is $\\Z \\xrightarrow{2} \\Z \\xrightarrow{0} \\Z$, which gives $H_0 = \\Z$, $H_1 = \\Z/2$ and $H_2 = 0$.\n\nNon-example: the definition needs a CW structure, that is, a filtration by skeleta in which each $X^n$ comes from $X^{n-1}$ by attaching $n$-cells.\nFor an arbitrary increasing chain of subspaces, the groups $H_n(X^n, X^{n-1})$ need not vanish outside degree $n$, and the homology of the resulting complex need not be the homology of $X$.\n",
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