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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $f \\colon S^n \\to S^n$ be a [[continuous map|0VGFHTM75Y]] of the [[$n$-sphere|T62369RFH6]].\nThe induced homomorphism $f_\\ast \\colon H_n(S^n; \\Z) \\to H_n(S^n; \\Z)$ on [[singular homology|M0Q5CFJ7ZH]] is an endomorphism of a group isomorphic to $\\Z$.\nSo $f_\\ast(a) = d \\, a$ for a unique integer $d$ and all $a$.\nThis integer is the *degree* of $f$, written $\\deg f$.\n\nMore generally, let $M$ and $N$ be closed connected [[oriented|913WEQERBE]] [[topological $n$-manifolds|QQQ9B1DK4P]] with [[fundamental classes|ZEA0PEM3B2]] $[M]$ and $[N]$, and let $f \\colon M \\to N$ be continuous.\nThe degree of $f$ is the integer $\\deg f$ with $f_\\ast [M] = (\\deg f) [N]$ in $H_n(N; \\Z)$.\n\n## Remarks\n\nHomotopic maps have the same degree, and $\\deg (g \\circ f) = \\deg g \\cdot \\deg f$.\n\nExample: on $S^1 \\subset \\C$, the map $f(z) = z^k$ has degree $k$.\nFor a reflection $r$ of $S^n$ in a hyperplane, $\\deg r = -1$; the antipodal map $x \\mapsto -x$ is a composite of $n + 1$ reflections, so its degree is $(-1)^{n+1}$.\n\nNon-example: a map $S^m \\to S^n$ with $m \\neq n$ has no degree in this sense, since the source and target manifolds must have the same dimension.\n\nIf $f$ is smooth and $y$ is a regular value, then $\\deg f = \\sum_{x \\in f^{-1}(y)} \\sgn \\det Df(x)$, where the sign is computed in oriented charts.\nFor $n = 1$, the degree of a map $S^1 \\to S^1$ equals its [[winding number|0NQSV2EWDR]] about $0$.\n",
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