Where is my logic wrong in the paragraph below?

Consider the mobius band in the context of differential geometry. It isn't orientable so there must be two chart transition maps from a differentiable atlas for the mobius band such that they have differing sign for the Jacobian determinant. But the determinant is a continuous map and since its range in this case includes $0$ there must be a point $p=det^{-1}(0)$ on the mobius band such that in a nbhd of $p$ any two charts with intersecting domain are not admissible to the atlas for the mobius band. This would mean any atlas for a mobius band can not completely cover it, right?

EDIT: Sorry, to the people that read this prior to my edits. But, I believe I've formulated my question correctly at this point and hopefully there will be no more edits.

  • $\begingroup$ $f:(-3,-1)\cup(1,3)\to \mathbb{R},x\to x$ is continuous. $f(-2)<0,f(2)>0$ and however $f$ doesn't take the value $0.$ Do you see the reason? $\endgroup$ – mfl Nov 28 '16 at 21:00
  • $\begingroup$ The domain of $f$ isn't connected. And as Andrew pointed out, in the case of overlapping charts they need not be. $\endgroup$ – Bob Nov 28 '16 at 21:09
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    $\begingroup$ Exactly that's the reason. But even assuming each overlap is connected, the problem is that if the manifold is non-orientable then you can't make any Jacobian positive. If you change a chart to make it positive then some other overlap becomes negative. As @Bob says, non-orientability is not a local property. $\endgroup$ – mfl Nov 28 '16 at 21:14
  • $\begingroup$ Do you mean positive everywhere? $\endgroup$ – Bob Nov 28 '16 at 21:16
  • $\begingroup$ I mean all overlap Jacobians positive. $\endgroup$ – mfl Nov 28 '16 at 21:18

The overlap of two charts need not be connected; the determinant of the overlap can be positive on one component and negative on another without vanishing. If the Möbius band is covered by two charts, the overlap cannot be connected (by your argument).

On the other hand, if the Möbius band is covered by three or more charts, then each overlap of two charts can be connected, but the overlap of a single chart with its "neighbors" needn't be connected.

As Jeff Weeks puts it in The Shape of Space, non-orientability is not a local property.

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    $\begingroup$ Thanks, yes, it just seemed to me that the domain of the intersection of charts must be connected but that isn't actually included in any treatment of manifold atlases that I've seen. And why should it be included? After all, differentiability is a local property. $\endgroup$ – Bob Nov 28 '16 at 21:11

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