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I know that one can turn a sphere $S^2$ in $\mathbb{R}^3$ inside-out having at each time an immersion of $S^2$ into $\mathbb{R}^3$. It is called Smale paradox. There is beautiful animation about that.

Do you know if the same thing is possible for higher dimensional spheres? I mean, for a sphere $S^k$ in $\mathbb{R}^{k+1}$ for $k\geq 3$.

Thank you!

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Hirsch extended Smale’s theorem, his extension is called the Smale Hirsch Theorem. It applies to your question, and says that the map that sends an immersion $S^n \to \mathbb R^{n+1}$ to its derivative $TS^n \to T\mathbb R^{n+1}$. is a homotopy-equivalence. Specifically, you consider the domain of this map to be the *space of immersions* of the sphere in Euclidean space, and the range to be the space of *bundle monomorphisms* from the tangent bundle of the sphere to the tangent bundle of Euclidean space.

Using the “cross product” construction you can convert between bundle monomorphisms $TS^n \to T\mathbb R^{n+1}$ and maps $S^n \to SO_{n+1}$. So the question of whether or not a sphere can be turned inside-out converts to whether or not these two maps $S^n \to SO_{n+1}$ (for the standard sphere and mirror reflection of the standard sphere) are homotopic. So when $n=2$ we’re done, since $\pi_2 SO_3 = 0$, but these maps can be non-trivial in higher dimensions.

First off, what are these elements of $\pi_n SO_{n+1}$? They come from the standard immersion via this cross-product construction I mentioned. For the standard immersion of the sphere, the element of $\pi_n SO_{n+1}$ constructed is the trivial element. How about for the mirror-reflected immersion? If you think about it a little, you see it is a composite of two mirror reflections, first the mirror reflection across a fixed vector and 2nd the mirror reflection across the plane orthogonal to the point in $S^n$. This map $S^n \to SO_{n+1}$ is known as the “clutching map” for $TS^{n+1}$.

So the answer is, you can turn $S^n$ inside-out in $\mathbb R^{n+1}$ if and only if the tangent bundle for the next higher-dimensional sphere $S^{n+1}$ is trivial. Since $S^0$, $S^1$, $S^3$ and $S^7$ are the only spheres with trivial tangent bundles, that answers your question. i.e. you can only turn $S^0$, $S^2$, and $S^6$ inside-out.

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