There's this problem in Lee's Intro to smooth manifold (5-2) that asks to prove that the Möbius bundle is a line bundle over the circle and that it is non trivial.(adsbygoogle = window.adsbygoogle || []).push({});

The Möbius bundle is [0,1]xR/~ where ~ identifies (0,y) with (1,-y) and the projection map p send [(x,y)] down to exp{(2pi)ix}.

So if the Möbius bundle were trivial, we would have that [0,1]xR/~ is homeomorphic toS^1 xR. (the open Möbius strip homeomorphic to the open cylinder)

The thing is that this problem comes before the chapter on orientation. So either there is a way to prove that the open Möbius strip is not homeomorphic to the open cylinder without invoking orientations, or there is some other way to show that the Möbius fribration is not trivial...

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# Nontriviality of the Möbius bundle without orientability

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