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  "body": "\n## Statement\n\nLet $X$ and $Y$ be connected [[CW complexes|5W95QFHXPR]], let $x_0 \\in X$, and let $f \\colon X \\to Y$ be a continuous map.\nAssume that $f_* \\colon \\pi_n(X, x_0) \\to \\pi_n(Y, f(x_0))$ is an isomorphism of [[homotopy groups|P976WYEFMB]] for every $n \\ge 1$.\nThen $f$ is a [[homotopy equivalence|XZG6MTSWCN]].\n\n## Remarks\n\nReplace $Y$ by the mapping cylinder of $f$, so that $f$ becomes the inclusion of a subcomplex.\nThe relative homotopy groups of the pair then vanish, and one deforms $Y$ onto $X$ cell by cell, using compression of maps of disks.\nIn other words, a [[weak homotopy equivalence|4Q3WY1RX91]] between CW complexes is a homotopy equivalence.\nThe CW hypothesis is needed: the map from the [[Warsaw circle|17VG475BRV]] to a point induces isomorphisms on all homotopy groups, but it is not a homotopy equivalence.\nThe isomorphisms must be induced by one map: $\\R P^2 \\times S^3$ and $S^2 \\times \\R P^3$ have isomorphic homotopy groups, but they are not homotopy equivalent, as their homology differs.\n\n## Explanations\n\n### L1 human\n\nHomotopy groups record how spheres of each dimension sit in a space.\nFor spaces built from cells, a map that matches all these records perfectly is as good as an equivalence: it can be inverted up to deformation.\nSo for such spaces, homotopy groups detect sameness of shape, provided a map realises the match.\n\n### L2 human\n\nThe theorem justifies working with CW complexes in homotopy theory, where weak and genuine homotopy equivalences coincide.\nCombined with the [[Hurewicz theorem|ZA11N5XPZM]], it gives a homology version.\nA map between simply connected CW complexes that induces isomorphisms on all integral homology groups is a homotopy equivalence.\nFor example, a simply connected CW complex with the integral homology of a point is contractible.\n",
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