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  "body": "\n## Summary\n\nThe Euler class $e(E) \\in H^n(B; \\Z)$ is a [[characteristic class|7HACJC0JE2]] of oriented real vector bundles $E \\to B$ of rank $n$.\nIt is the first obstruction to a nowhere-zero section: if $E$ has a section that never vanishes, then $e(E) = 0$.\nFor the tangent bundle of a closed oriented manifold, its value on the [[fundamental class|ZEA0PEM3B2]] is the [[Euler characteristic|RNDDWDAFJ5]].\nIt appears in the [[Gysin sequence|QJSPWM9F9P]] and is defined from the Thom class of the [[Thom isomorphism|N685RXHDZX]].\n\n## Explanations\n\n### L1 human\n\nThe Euler class counts, with signs, the zeros that a section of a bundle is forced to have.\nFor the tangent bundle of a surface, a section is a field of arrows, and the count is the Euler characteristic of the surface.\nOn the sphere the count is $2$, so every field of arrows on the sphere has a zero somewhere; this is the hairy ball theorem.\nOn the torus the count is $0$, and indeed the torus carries a field of arrows with no zeros.\n",
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