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  "body": "\n## Definition\n\nLet $n \\ge 2$, and let $f \\colon S^{2n-1} \\to S^n$ be a continuous map.\nLet $C_f = S^n \\cup_f D^{2n}$ be the [[CW complex|5W95QFHXPR]] obtained by attaching a $2n$-cell to $S^n$ along $f$.\nIts [[singular cohomology|6N7BC70CX0]] with integer coefficients is $H^k(C_f; \\Z) \\cong \\Z$ for $k = 0, n, 2n$ and is zero otherwise.\nLet $\\alpha \\in H^n(C_f; \\Z)$ and $\\beta \\in H^{2n}(C_f; \\Z)$ be the generators given by fixed orientations of $S^n$ and $D^{2n}$.\nThe *Hopf invariant* of $f$ is the integer $H(f)$ such that\n$$\\alpha \\smile \\alpha = H(f) \\, \\beta,$$\nwhere $\\smile$ is the [[cup product|JX47MS1R1H]].\n\n## Remarks\n\nThe integer $H(f)$ depends only on the homotopy class of $f$, and $H$ is a homomorphism $\\pi_{2n-1}(S^n) \\to \\Z$ on [[homotopy groups|P976WYEFMB]].\nA change of the generator $\\beta$ changes only the sign of $H(f)$.\n\nExample: for the Hopf map $h \\colon S^3 \\to S^2$, the space $C_h$ is $\\C P^2$.\nThe square of a generator of $H^2(\\C P^2; \\Z)$ generates $H^4(\\C P^2; \\Z)$, so $H(h) = \\pm 1$.\n\nNon-example: for $n = 1$ the construction breaks down.\nA map $S^1 \\to S^1$ of degree $d$ gives $H^2(C_f; \\Z) \\cong \\Z/d$, which is not $\\Z$ when $d \\ne 0$, so no generator $\\beta$ as above exists.\n\nIf $n$ is odd, then $\\alpha \\smile \\alpha = -\\alpha \\smile \\alpha$ by graded commutativity, and since $H^{2n}(C_f; \\Z) \\cong \\Z$ has no torsion, $H(f) = 0$.\nIf $n$ is even, the Whitehead product $[\\iota_n, \\iota_n]$ of the identity of $S^n$ with itself has Hopf invariant $\\pm 2$.\n",
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