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Sphere eversion is the process of turning a sphere inside out in a three-dimensional space. See also this animation on YouTube: Outside in (2011) and picture below. My question is, what is the historical context behind the problem? More specifically, why was the problem raised?

enter image description here

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    $\begingroup$ Did you read the history section of that article? Smale characterized regular homotopy classes of immersions of $2$-spheres (homotopy classifications were a standard fare in 1930-60s), from which existence of an eversion followed. Bott thought it wrong (mistakenly), and even when the mistake was sorted out others still found it counterintuitive. Before Smale, "there was no documented attempt to argue for or against the eversion of $S^2$, and later efforts are in hindsight, so there never was a historical paradox associated with sphere eversion". $\endgroup$
    – Conifold
    Aug 22, 2023 at 5:56

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