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  "body": "\n## Summary\n\nAn orientation of an $n$-dimensional [[manifold|QQQ9B1DK4P]] is a consistent choice, at every point, of one of the two local directions of rotation, or \"handedness\".\nIn algebraic topology it is a continuous choice of generator of the local homology group $H_n(M, M \\setminus \\set{x}) \\cong \\Z$ at each point $x$.\nFor smooth manifolds it is the same as a continuous choice of [[orientation|0QJJAQ6ZYZ]] of every tangent space.\nOrientations are needed for fundamental classes, Poincaré duality with integer coefficients, and integration of differential forms.\n\n## Explanations\n\n### L1 human\n\nOn a small piece of surface you can always decide which way is clockwise.\nA surface is orientable if you can make this decision everywhere so that nearby choices agree.\nA sphere is orientable: clockwise as seen from outside works everywhere.\nA Möbius band is not: carry a small clockwise circle once around the band and it comes back counterclockwise.\nAn [[orientable surface|0TD904MG06]] is a surface where such a global choice exists.\n",
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