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  "body": "\n## Summary\n\nA manifold is orientable when one can choose an orientation at every point, varying continuously over the whole manifold.\nAn orientation is a sense of \"clockwise\" in dimension $2$, or of \"right-handed\" in dimension $3$.\nThe sphere and the torus are orientable; the Möbius strip, the Klein bottle and the real projective plane are not.\nOrientability is needed to integrate differential forms of top degree, to define the fundamental class and to state Poincaré duality with integer coefficients.\n\n## Explanations\n\n### L1 human\n\nDraw a small clockwise circle on a surface and slide it around along any closed path.\nOn a sphere, the circle always comes back clockwise.\nOn a [[Möbius strip|04AJE0M271]], one trip around the strip brings it back turning the other way.\nA surface is orientable when such a reversal never happens, so that \"clockwise\" has one meaning everywhere.\nIn the same way, a $3$-dimensional manifold is orientable when a right-handed glove can never be carried around a loop and come back as a left-handed one.\n",
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