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I heard an easy way to see if a connected 2-dimensional manifold is non-orientable is applying local rotations (clockwise) in a consistant way. And if you encounter an anti-clockwise rotation the manifold can't be orientable. I tried to apply this on the Möbius Band but i haven't encountered ani anti-clockwise rotations, everything is consistent. This shouldn't be the case. And i'm 99% sure i have a proper Möbius Band. Is this method wrong in this case or am i doing something wrong here?

Thanks for your help!

Edit: I guess i found my mistake, i locally treated the paper's front- and backside different. If i just follow the band and apply local rotations - of course i'll met the same rotation at some point. I just had to compare the consistency of a rotation on the frontside with the rotaton on the backside. Well...

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You must be doing something wrong. What, we can't tell without seeing what you did and have. –  Daniel Fischer Sep 26 '13 at 15:45
Your edit explains things correctly. I've seen the following idea mentioned as well. Imagine you have a hole puncher that punches out an asymmetric shape of some kind. If you keep applying this hole punch as you work your way around the Möbius strip, eventually the hole you cut out meets its mirror image. My wife made a Möbius quilt that illustrates this idea. –  Grumpy Parsnip Sep 26 '13 at 19:14
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2 Answers

Draw the Möbius band as a rectangle with the usual identifications on the top and bottom edge and triangulate the space by first splitting the rectangle in to two squares, one on top of the other, and then splitting these squares diagonally in to right angled triangles.

Put an orientation on one of the 2-simplices. It will induce an orientation on all of the other simplices except for one, as the last simplex will have two simplices sharing an edge with it, and their induced orientations will not coincide. (You should draw this to check).

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This is the best way I can think of explaining it:

An orientation is a locally consistent choice of ordering on the bases for the tangent spaces. With a Möbius strip, if you suppose the tangent space at a point had an orientation (say 'clockwise'), you're forced to continue with the 'clockwise' orientation along the strip. Before you get back to your initial point, however, the band will have twisted, so you have an anti-clockwise orientation where you started with a clockwise orientation. The Möbius band is thin, so this is the problem.

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