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  "body": "\n## Definition\n\nLet $X$ be a topological space and let $A \\subseteq X$ be a subspace with the [[subspace topology|0WS9P12FQR]].\nLet $C_\\bullet(X)$ be the singular chain complex of $X$.\nEvery singular simplex in $A$ is a singular simplex in $X$, so $C_\\bullet(A)$ is a subcomplex of $C_\\bullet(X)$.\nThe *relative chain complex* is the quotient $C_n(X, A) = C_n(X) / C_n(A)$, with the boundary map induced by the boundary of $C_\\bullet(X)$.\nThe *relative homology* $H_n(X, A)$ is the homology of $C_\\bullet(X, A)$ in degree $n$.\n\n## Remarks\n\nA class in $H_n(X, A)$ is represented by a *relative cycle*, a chain $c \\in C_n(X)$ with $\\partial c \\in C_{n-1}(A)$.\nTwo relative cycles give the same class when their difference is $\\partial b + a$ with $b \\in C_{n+1}(X)$ and $a \\in C_n(A)$.\nFor $A = \\emptyset$, the group $H_n(X, \\emptyset)$ is the [[singular homology|M0Q5CFJ7ZH]] $H_n(X)$.\nA map of pairs $f \\colon (X, A) \\to (Y, B)$, that is, a continuous map with $f(A) \\subseteq B$, induces maps $f_* \\colon H_n(X, A) \\to H_n(Y, B)$.\n\nExample: for $n \\ge 1$, $H_n(D^n, S^{n-1}) \\cong \\Z$ and the other relative groups vanish.\nFor $n = 1$, the path $\\sigma(t) = 2t - 1$ in $D^1 = [-1, 1]$ is a relative cycle, since $\\partial \\sigma = (1) - (-1)$ lies in $C_0(S^0)$.\n\nNon-example: in $X = [-1, 1]$ with $A = \\set{-1}$, the same path is not a relative cycle, because its boundary contains the point $1 \\notin A$.\n",
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