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  "body": "\n## Definition\n\nLet $K$ be a [[simplicial complex|V8GZ0W1ZZ5]] with a total order on its vertex set.\nFor $n \\ge 0$ let $C_n(K)$ be the [[free abelian group|0WJX8MGG37]] with basis the $n$-simplices of $K$, and let $C_n(K) = 0$ for $n \u003c 0$.\nAn $n$-simplex with vertices $v_0 \u003c v_1 \u003c \\cdots \u003c v_n$ is written $[v_0, \\dots, v_n]$.\nThe *boundary map* $\\partial_n \\colon C_n(K) \\to C_{n-1}(K)$ is the homomorphism given on basis elements by\n$$\\partial_n [v_0, \\dots, v_n] = \\sum_{i=0}^{n} (-1)^i [v_0, \\dots, \\widehat{v_i}, \\dots, v_n],$$\nwhere $\\widehat{v_i}$ means that $v_i$ is omitted, and $\\partial_0 = 0$.\nThen $\\partial_{n} \\circ \\partial_{n+1} = 0$, so $(C_\\bullet(K), \\partial)$ is a chain complex.\nThe *$n$-th simplicial homology group* of $K$ is\n$$H_n(K) = \\ker \\partial_n / \\operatorname{im} \\partial_{n+1}.$$\n\n## Remarks\n\nElements of $\\ker \\partial_n$ are $n$-cycles and elements of $\\operatorname{im} \\partial_{n+1}$ are $n$-boundaries; the [[kernel|0Z424HK8CQ]] contains the image because $\\partial_n \\circ \\partial_{n+1} = 0$.\nThe groups do not depend on the vertex order up to isomorphism.\nWith coefficients in an abelian group $A$, one uses $C_n(K) \\otimes A$.\n\nExample: let $K$ be the boundary of a triangle, with vertices $0 \u003c 1 \u003c 2$ and edges $[0,1]$, $[0,2]$, $[1,2]$.\nThen $\\partial_1 [0,1] = [1] - [0]$, $\\partial_1 [0,2] = [2] - [0]$, $\\partial_1 [1,2] = [2] - [1]$.\nThe kernel of $\\partial_1$ is generated by $[0,1] - [0,2] + [1,2]$, and there are no $2$-simplices, so $H_1(K) \\cong \\Z$; also $H_0(K) \\cong \\Z$.\n\nNon-example: dropping the signs $(-1)^i$ does not give a chain complex over $\\Z$.\nFor the $2$-simplex, the unsigned boundary of $[0,1,2]$ is $[1,2] + [0,2] + [0,1]$, whose unsigned boundary is $2([0] + [1] + [2]) \\ne 0$.\n",
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