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  "body": "\n## Definition\n\nObstruction theory is built on the obstruction cocycle, defined as follows.\nLet $(X, A)$ be a [[relative CW complex|D0DAFG8QK6]], and let $X_k$ be the union of $A$ with the cells of $X$ of dimension at most $k$.\nLet $n \\ge 1$ and let $Y$ be a path-connected space whose fundamental group acts as the identity on the [[homotopy group|P976WYEFMB]] $\\pi_n(Y)$.\nThen free homotopy classes of maps $S^n \\to Y$ correspond to elements of $\\pi_n(Y)$.\nLet $g \\colon X_n \\to Y$ be a map.\nFix an orientation of each $(n+1)$-cell $e$ of $X \\setminus A$, with attaching map $\\varphi_e \\colon S^n \\to X_n$.\nThe *obstruction cocycle* of $g$ is the cellular cochain\n$$c(g) \\in C^{n+1}(X, A; \\pi_n(Y)), \\qquad c(g)(e) = [g \\circ \\varphi_e].$$\nIts class $[c(g)]$ lies in $H^{n+1}(X, A; \\pi_n(Y))$, which is isomorphic to the cellular cohomology and to [[singular cohomology|6N7BC70CX0]] of the pair.\n\n## Remarks\n\nThe basic facts of obstruction theory are these.\n\n1. The cochain $c(g)$ is a cocycle.\n2. The map $g$ extends to $X_{n+1}$ if and only if $c(g) = 0$.\n3. The class $[c(g)]$ is zero if and only if the restriction of $g$ to $X_{n-1}$ extends to a map $X_{n+1} \\to Y$.\n\nFact 2 holds because a map $S^n \\to Y$ extends over $D^{n+1}$ exactly when it is null-homotopic.\nFact 3 is proved with the difference cochain of two maps that agree on $X_{n-1}$.\n\nExample: let $(X, A) = (D^{n+1}, S^n)$, where $X \\setminus A$ is a single $(n+1)$-cell, so $X_n = S^n$.\nLet $n \\ge 1$, $Y = S^n$, and let $g$ be the identity of $S^n$.\nThen $c(g)$ sends the top cell to the generator of $\\pi_n(S^n) \\cong \\Z$, so $g$ does not extend; this is the no-retraction theorem.\n\nNon-example: if $\\pi_1(Y)$ acts nontrivially on $\\pi_n(Y)$, the value $[g \\circ \\varphi_e]$ is only defined up to this action, so $c(g)$ is not defined with constant coefficients; one then uses local coefficients.\n",
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