I'm trying to wrap my head around the concept of orientability as an intrinsic property of a manifold. Assume I'm in some (3-dim) manifold for which I'd like to decide its orientability; what could I do in that case? I guess this setting is sufficient to avoid thinking with immersions/embeddings.

My attempts so far:

  • Comparing to the 2D case. If a 2D object travels along the Möbius strip, it would appear flipped (left and right) when it's back to its original location.

    But that still sounds abstract. What would I perceive if I travel around in a 3-manifold and come back? Maybe things suddenly look mirrored, I guess, but not sure. And I see no reason why it has to be left-right (instead of upside-down, since that's also perpendicular to my travel curve).

  • I did find some hints in another question; the above is similar to the idea of the second answer there. Another idea is to observe counter-clockwise-ness, mentioned in the comments of the first answer. Neither of them is sufficiently detailed to make me understand, so I have to ask this separate question.

Thanks in advance.

  • $\begingroup$ There are round-trips after which everything looks mirrored. But in general, in a differentiable 3-manifold, there is no way for you to decide whether a left-right-flip or an up-down-flip occurred, because there is no canonical way for you to fix an up-direction during your travel. This might change when you add a connection (I am no expert, so let's wait for other "opinions"). $\endgroup$ – M. Winter Apr 25 '18 at 7:33
  • $\begingroup$ the straight answer: the three manifold ought to have a solid Klein-bottle or not. To get some intuition on the solid Klein-bottle one studies the cartesian product $M\dot{o}\times I$, of the Möbius band by an interval. $\endgroup$ – janmarqz Apr 25 '18 at 22:36
  • $\begingroup$ If our space where NOT orientable, you could not make any difference between your left and your right hand. A manifold is NOT orientable if after a certain trip in this manifold, and coming back to your origibnal position your left hand is what your left hand was. $\endgroup$ – Thomas Apr 26 '18 at 10:31

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