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  "body": "\n## Definition\n\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 an [[$(n-1)$-connected|1A943BWBQT]] space; if $n = 1$, assume also that $\\pi_1(Y)$ is abelian.\nLet $f \\colon A \\to Y$ be a map.\nSince $\\pi_k(Y) = 0$ for $k \u003c n$, the map $f$ extends to a map $g \\colon X_n \\to Y$.\nThe *primary obstruction* of $f$ is the class\n$$\\gamma(f) = [c(g)] \\in H^{n+1}(X, A; \\pi_n(Y)),$$\nwhere $c(g)$ is the obstruction cocycle of $g$ from [[obstruction theory|ZZWHGGFB4P]].\n\n## Remarks\n\nThe class $\\gamma(f)$ does not depend on the choice of $g$.\nIndeed, any two extensions of $f$ to $X_n$ are homotopic relative to $A$ on $X_{n-1}$, since $Y$ is $(n-1)$-connected.\nThe difference of the two extensions on the $n$-cells changes $c(g)$ by a coboundary.\nThe map $f$ extends to $X_{n+1}$ if and only if $\\gamma(f) = 0$.\nIf $X \\setminus A$ has no cells of dimension above $n + 1$, then $f$ extends to $X$ if and only if $\\gamma(f) = 0$.\n\nExample: let $(X, A) = (D^{n+1}, S^n)$, $Y = S^n$ with $n \\ge 1$, and $f$ the identity.\nHere $H^{n+1}(D^{n+1}, S^n; \\pi_n(S^n)) \\cong \\Z$ and $\\gamma(f)$ is a generator, so $f$ does not extend.\n\nNon-example: let $Y$ be not $(n-1)$-connected, for instance $Y = S^1$ with $n = 2$.\nThen $f$ need not extend over $X_2$ at all, so no class in degree $n + 1$ is defined from $f$ alone.\n\nThe same construction applies to a fibration over a CW complex with $(n-1)$-connected fibre $F$.\nIt gives the primary obstruction to a section, a class in $H^{n+1}$ of the base with coefficients in $\\pi_n(F)$, local in general.\n",
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