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  "body": "\n## Statement\n\nLet $\\pi \\colon E \\to B$ be an oriented real vector bundle of rank $n \\ge 1$, let $E_0$ be the complement of its zero section, and let $\\pi_0 = \\pi|_{E_0} \\colon E_0 \\to B$.\nLet $e \\in H^n(B; \\Z)$ be its [[Euler class|BJX6QS0ANT]].\nThen there is a long [[exact sequence|JB7ZBFBA9C]] in integral [[singular cohomology|6N7BC70CX0]]\n$$\\cdots \\to H^i(B) \\xrightarrow{\\ \\smile e\\ } H^{i+n}(B) \\xrightarrow{\\ \\pi_0^*\\ } H^{i+n}(E_0) \\to H^{i+1}(B) \\xrightarrow{\\ \\smile e\\ } H^{i+n+1}(B) \\to \\cdots,$$\nwhere $\\smile e$ is the [[cup product|JX47MS1R1H]] with $e$.\n\n## Remarks\n\nThe space $E_0$ deformation retracts onto the unit [[sphere bundle|3VQ2V5HH3R]] $S(E)$ of any metric on $E$, so the sequence is often written with $S(E)$, a bundle with fibre $S^{n-1}$.\nThe proof writes the long exact sequence of the pair $(E, E_0)$, replaces $H^\\bullet(E)$ by $H^\\bullet(B)$ since $E$ deformation retracts onto $B$, and replaces $H^{\\bullet}(E, E_0)$ by $H^{\\bullet - n}(B)$ with the [[Thom isomorphism|N685RXHDZX]].\nThe composite $H^{i}(B) \\to H^{i+n}(E, E_0) \\to H^{i+n}(E) \\cong H^{i+n}(B)$ is then the cup product with $e$.\nWith $\\Z/2$ coefficients the sequence holds without the orientation hypothesis, with the top Stiefel–Whitney class in place of $e$.\n\nExample: the tautological complex line bundle over complex projective space $\\C P^m$ has rank $2$ as a real bundle, and its unit sphere bundle is the Hopf map $S^{2m+1} \\to \\C P^m$.\nSince $H^j(S^{2m+1}) = 0$ for $0 \u003c j \u003c 2m+1$, the sequence shows that $\\smile e \\colon H^i(\\C P^m) \\to H^{i+2}(\\C P^m)$ is an isomorphism for $0 \\le i \\le 2m - 2$.\nWith $H^0(\\C P^m) = \\Z$, this gives $H^\\bullet(\\C P^m; \\Z) \\cong \\Z[e]/(e^{m+1})$.\n\n## Explanations\n\n### L1 human\n\nA sphere bundle is a space made of spheres, one over each point of a base space.\nThe Gysin sequence computes the cohomology of the total space from the cohomology of the base and one class, the Euler class, which measures how twisted the bundle is.\nIf the bundle is a product, the Euler class is zero and the sequence splits into short pieces.\n\n### L2 human\n\nThe Gysin sequence is the special case of the Serre spectral sequence for a sphere bundle, where only two rows are nonzero.\nIt is a standard way to compute cohomology rings of projective spaces, of unit tangent bundles and of lens spaces.\nIt also shows that an oriented bundle with a nowhere zero section has Euler class zero, because then $\\pi_0^*$ is injective in the relevant degree.\n",
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