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I saw this YouTube video claiming that spheres, double torues, triple toruses, etc. can all be turned inside out, but what about other surfaces? Are there any surfaces that can't be turned inside out?

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In mathematical terms, your question may be restated as "does every closed orientable manifold of dimension two admit an orientation reversing diffeomorphism?" As noted by others, the question does not make sense for non orientable surfaces, and so we have to limit ourselves to orientable spaces.

A key phrase here is the classification of surfaces, which puts surfaces into three categories: the sphere, tori with several holes (surfaces of genus $g$), and several copies of the projective plane glued together. It can be shown that any closed $2$-manifold must be of this type. So in short yes, since the orientable surfaces are precisely the tori with several holes, all orientable surfaces admit an orientation reversing diffeomorphism.

It should be noted that the answer is negative for higher dimensional orientable spaces. For example, even dimensional complex projective spaces do not admit orientation reversing diffeomorphisms.

Edit: I am limiting myself here to closed orientable surfaces. Non-closed surfaces are not as nice and I'm not sure there's a clear answer, perhaps someone else can fill in those details.

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    $\begingroup$ To me "can be turned inside out" suggests something much stronger than the existence of an orientation-reversing diffeomorphism - indeed, it's obvious that the antipodal map of the sphere reverses orientation, but the sphere eversion theorem was a great surprise. $\endgroup$ – Anthony Carapetis Oct 15 '17 at 4:57
  • $\begingroup$ Interesting. How does one formalize the idea in general? $\endgroup$ – leibnewtz Oct 15 '17 at 5:11
  • $\begingroup$ The sphere eversion theorem seems to be a bit more general than just the claim that I can be turned inside out. To me it seems to be saying that it can be turned inside out, regardless of how you embed it. But I assume the OP cares only about the standard embedding $\endgroup$ – leibnewtz Oct 15 '17 at 5:18
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    $\begingroup$ Well, it's about being able to smoothly deform the standard embedding to one with the same image, but the orientation reversed. This is formalized by requiring the two embeddings to be homotopic through immersions. Given that the OP linked a video specifically talking about Smale's result, I don't think orientation-preserving diffeomorphisms alone answer the question. $\endgroup$ – Anthony Carapetis Oct 15 '17 at 7:30
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A Klein bottle is a surface with no inside and no outside. So in that case I don't think it would make any sense to talk about turning it inside out.

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For the class of surfaces called non-orientable, there's no such thing as inside or out.

This class includes the famous Moebius Strip and Klein Bottle, for instance.

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  • $\begingroup$ Ok, but what about surfaces that do have an inside? $\endgroup$ – user80458 Oct 15 '17 at 1:54
  • $\begingroup$ This question deserves a better answer than this. $\endgroup$ – MJD Oct 15 '17 at 2:26
  • $\begingroup$ I agree, @mjd. If I have time later I'll expand, in the meantime why don't you give it a shot? $\endgroup$ – dbx Oct 15 '17 at 12:10

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