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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $M$ be a compact connected [[oriented|913WEQERBE]] [[smooth manifold|F45VQ47GPV]] of dimension $n$ without boundary.\nLet $f, g \\colon M \\to S^n$ be continuous maps to the [[$n$-sphere|T62369RFH6]].\nThen $f$ and $g$ are [[homotopic|F2F143KQ2M]] if and only if $\\deg f = \\deg g$.\nMoreover, every integer is the [[degree|TMFG2P912R]] of some continuous map $M \\to S^n$.\n\n## Remarks\n\nOne direction is the homotopy invariance of degree.\nFor the other, one may take $f$ and $g$ smooth and use a regular value.\nBy the Pontryagin–Thom construction, the homotopy class of a map $M \\to S^n$ is determined by the framed cobordism class of the preimage of a regular value.\nThis preimage is a finite set of signed points, and in dimension $0$ such a class is determined by the sum of the signs.\nIn other words, the degree gives a bijection from the set $[M, S^n]$ of homotopy classes to $\\Z$.\n\nExample: for $M = S^1$ and $n = 1$, the theorem says that loops in the circle are classified up to homotopy by their winding number, which recovers $\\pi_1(S^1) \\cong \\Z$.\n\nThe dimensions must agree: the Hopf map $S^3 \\to S^2$ is not null-homotopic, though $H_3(S^2; \\Z) = 0$ so that no degree can distinguish it from a constant map.\nIf $M$ is not orientable, the same holds with the degree replaced by the degree modulo $2$.\n\n## Explanations\n\n### L1 human\n\nTake two maps from a closed oriented manifold to a sphere of the same dimension, for example from a closed surface to $S^2$.\nOne can be deformed into the other exactly when both wrap around the sphere the same number of times.\nSo a single integer tells all about such maps up to deformation.\n\n### L2 human\n\nThe theorem computes $[M, S^n] \\cong \\Z \\cong H^n(M; \\Z)$ for closed connected oriented $n$-manifolds.\nIt is a first case of the representability of cohomology by Eilenberg–MacLane spaces.\nThe reason is that a map $S^n \\to K(\\Z, n)$ representing a generator induces isomorphisms on $\\pi_i$ for $i \\le n$, which is enough for maps out of an $n$-dimensional $M$.\nIt is used to define indices of vector fields and to prove the Poincaré–Hopf theorem, and it is the base case of obstruction theory for maps into spheres.\n",
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