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  "body": "\n## Definition\n\nLet $\\pi \\colon E \\to B$ be an oriented real vector bundle of rank $n$ over a paracompact space $B$.\nLet $s \\colon B \\to E$ be the zero section and let $E_0 \\subset E$ be its complement.\nLet $u \\in H^n(E, E_0; \\Z)$ be the Thom class of $E$.\nIt is the unique relative class whose restriction to $H^n(E_b, E_b \\setminus \\set{0}; \\Z) \\cong \\Z$ is the oriented generator, for every $b \\in B$.\nIts existence and uniqueness is part of the [[Thom isomorphism|N685RXHDZX]] theorem.\nThe *Euler class* of $E$ is\n$$e(E) = s^* j^* u \\in H^n(B; \\Z),$$\nwhere $j^* \\colon H^n(E, E_0; \\Z) \\to H^n(E; \\Z)$ is restriction to absolute [[cohomology|6N7BC70CX0]].\n\n## Remarks\n\nThe Euler class is natural under pullback, so it is a [[characteristic class|7HACJC0JE2]] of oriented bundles.\nReversing the orientation changes the sign of $e(E)$.\nIf $E$ has a nowhere-zero section $\\sigma$, then $s$ is homotopic to $\\sigma$ as maps into $E$ and $\\sigma$ factors through $E_0$, where $j^* u$ restricts to $0$; so $e(E) = 0$.\nIf $n$ is odd, then $2 e(E) = 0$, because the map $v \\mapsto -v$ reverses the orientation.\nThe reduction of $e(E)$ modulo $2$ is the top [[Stiefel–Whitney class|JQCK7A210W]] $w_n(E)$.\n\nExample: for a closed oriented manifold $M$, $\\langle e(TM), [M] \\rangle = \\chi(M)$.\nFor $M = S^2$ this gives $e(TS^2) = 2$ times the generator of $H^2(S^2; \\Z)$.\n\nNon-example: the Möbius line bundle over $S^1$ is not orientable, so it has no Euler class with integer coefficients in this sense; only its mod $2$ analogue $w_1$ is defined.\n",
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