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  "body": "\n## Statement\n\nLet $X$ be a [[path-connected|0WB6DH499T]] space with base point $x_0$, and let $n \\ge 1$.\nFix a generator $u_n$ of $H_n(S^n; \\Z) \\cong \\Z$.\nThe *Hurewicz map* $h \\colon \\pi_n(X, x_0) \\to H_n(X; \\Z)$ sends the class of $f \\colon S^n \\to X$ to $f_*(u_n)$.\nHere $\\pi_n$ is the [[homotopy group|P976WYEFMB]] and $H_n$ is [[singular homology|M0Q5CFJ7ZH]].\n\n1. For $n = 1$, $h$ is surjective and its kernel is the commutator subgroup of the [[fundamental group|S1XK3HXHFX]]. Hence $H_1(X; \\Z)$ is the [[abelianisation|0AQZV0905N]] of $\\pi_1(X, x_0)$.\n2. For $n \\ge 2$, assume that $X$ is [[$(n-1)$-connected|1A943BWBQT]]. Then the [[reduced homology|NJQMWVK3QK]] $\\tilde H_k(X; \\Z)$ vanishes for $k \u003c n$, and $h \\colon \\pi_n(X, x_0) \\to H_n(X; \\Z)$ is an isomorphism.\n\n## Remarks\n\nThe map $h$ is a homomorphism, because a sum of maps $S^n \\to X$ factors through the pinch map $S^n \\to S^n \\vee S^n$.\nFor $n \\ge 2$, one replaces $X$ by a weakly equivalent CW complex with a single $0$-cell and no cells of dimensions $1$ to $n - 1$.\nThen both groups are computed from the $n$-cells and the attaching maps of the $(n+1)$-cells.\nExample: $S^n$ is $(n-1)$-connected and $H_n(S^n) \\cong \\Z$, so $\\pi_n(S^n) \\cong \\Z$, generated by the identity.\nThe isomorphism stops at the first nonzero degree: $S^2$ is $1$-connected, yet $\\pi_3(S^2) \\cong \\Z$ while $H_3(S^2) = 0$.\n\n## Explanations\n\n### L1 human\n\nHomotopy groups are hard to compute, and homology groups are much easier.\nThe theorem says that the two agree in the first dimension where something happens.\nIn dimension one, homology is the fundamental group made commutative.\n\n### L2 human\n\nThe theorem gives the first nonzero homotopy group of a simply connected space from its homology.\nWith the [[Whitehead theorem|6P1E2FJPSD]], it shows that a homology isomorphism between simply connected CW complexes is a homotopy equivalence.\nA relative version for pairs of spaces extends it further.\nIt is also the starting point for Serre's mod $\\mathcal{C}$ theory, which compares homotopy and homology modulo classes of abelian groups.\n",
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